tV    THE 

uf  ^difopia, 


Name  of  Book  and  Volume, 


2 

r^IT?-?. 


Division 
Range 
Shelf.  ......  . 


187 


ELEMENTS 


OF 


PHYSICAL  MANIPULATION. 


BY 

EDWARD  C.  PICKERING, 

T/iayer  Professor  of  Physics  in  the  Massachusetts  Institute  of  Technology 


NEW    YORK: 

PUBLISHED   BY   HURD   AND    HOUGHTON. 
El)c  ftfterfftrc 

1873. 


Entered  according  to  Act  of  Congress  in  the  year  1873,  by 

EDWARD  C.  PICKERING, 
In  the  Oflice  of  the  Librarian  of  Congress,  Washington. 


TO 


A 

|r4 


THE   FIRST    TO    PROPOSE    A   PHYSICAL    LABORATORY, 

THIS  WORK  IS  MOST  AFFECTIONATELY  INSCRIBED 

BY  HIS 
SINCERE  FRIEND     AND    PUPIL, 

THE   AUTHOR. 


PREFACE. 


THE  rapid  spread  of  the  Laboratory  System  of  teaching  Physics, 
both  in  this  country  and  abroad,  seems  to  render  imperative  the 
demand  for  a  special  text-book,  to  be  used  by  the  student.  To 
meet  this  want  the  present  work  has  been  prepared,  based  on 
the  experience  gained  in  the  Massachusetts  Institute  of  Technol- 
ogy during  the  past  four  years.  The  preliminary  chapter  is 
devoted  to  general  methods  of  investigation,  and  the  more  com- 
mon applications  of  the  mathematics  to  the  discussion  of  results. 
The  graphical  method  does  not  seem  to  have  attracted  the  atten- 
tion it  deserves ;  it  is  accordingly  compared  here  with  the 
analytical  method.  Some  new  developments  of  it  are  moreover 
inserted.  It  is  of  fundamental  importance  that  the  student  should 
clearly  understand  how  to  deal  with  his  observations,  and  reduce 
them,  and  that  he  should  be  familiar  with  the  various  kinds  of 
errors  present  in  all  physical  experiments.  A  short  description  is 
also  given  of  the  various  methods  of  measuring  distances,  time 
and  weights,  which,  in  fact,  form  the  basis  of  all  physical  inves- 
tigation. This  chapter  is  intended  as  the  ground-work  of  a  short 
course  of  lectures,  given  to  the  students  before  they  begin  their 
work  in  the  laboratory.  It  should  be  so  far  extended  by  the 
instructor,  as  10  render  them  familiar  with  the  general  principles 

on  which  all  physical    instruments  are  constructed,  thus  greatly 

(v) 


VI  PREFACE. 

aiding  them  when  they  have  occasion  to  devise  apparatus  for 
their  own  work. 

The  remainder  of  the  volume  is  devoted  to  a  series  of  experi- 
ments which  it  is  intended  that  the  student  shall  perform  in  the 
laboratory.  Each  experiment  is  divided  into  two  parts ;  the  first 
called  Apparatus,  giving  a  description  of  the  instruments  re- 
quired, and  designed  to  aid  the  instructor  in  preparing  the  labora- 
tory for  the  class.  The  student  should  read  this  over,  and  with  it 
the  second  part,  entitled  Experiment,  which  explains  in  detail 
what  he  is  to  do. 

Perhaps  the  greatest  advantage  to  be  derived  from  a  course  of 
physical  manipulation,  is  the  means  it  affords  of  teaching  a  student 
to  think  for  himself.  This  should  be  encouraged  by  allowing  him 
to  carry  out  any  ideas  that  may  occur  to  him,  and  so  far  as  possible 
devise  and  construct,  with  his  own  hands,  the  apparatus  needed. 
Many  such  investigations  are  suggested  in  connection  with  some  of 
the  experiments,  for  instance  Nos.  13,  37,  48,  69,  77,  93  and  others. 
To  aid  in  this  work,  a  room  adjoining  the  laboratory  should  be  fit- 
ted up  with  a  lathe  and  tools  for  working  in  metals  and  wood,  as 
most  excellent  results  may  sometimes  be  attained  at  very  small 
expense,  by  apparatus  thus  constructed  by  students. 

The  method  of  conducting  a  Physical  Laboratory,  for  which  this 
book  is  especially  designed,  and  which  has  been  in  daily  use  with 
entire  success  at  the  Institute,  is  as  follows.  Each  experiment  is 
assigned  to  a  table,  on  which  the  necessary  apparatus  is  kept,  and 
where  it  is  always  used.  A  board  called  an  indicator  is  hung  on 
the  wall  of  the  room,  and  carries  two  sets  of  cards  opposite  each 
other,  one  bearing  the  names  of  the  experiments,  the  other  those 
of  the  students.  When  the  class  enters  the  laboratory,  each 
member  goes  to  the  indicator,  sees  what  experiment  is  assigned 
to  him,  then  to  the  proper  table  where  he  finds  the  instruments 
required,  and  by  the  aid  of  the  book  performs  the  experiment. 


PREFACE.  Vli 

Any  additional  directions  needed  are  written  on  a  card  also  placed 
on  the  table.  As  soon  as  the  experiment  is  completed,  he  reports 
the  results  to  the  instructor,  who  furnishes  him  with  a  piece  of 
paper  divided  into  squares  if  a  curve  is  to  be  constructed,  or  with 
a  blank  to  be  filled  out,  when  single  measurements  only  have  been 
taken.  In  either  case  a  blank  form  is  supplied,  as  a  copy.  New 
work  is  then  assigned  to  him  by  merely  moving  his  card  opposite 
any  unoccupied  experiment.  By  following  this  plan  an  instructor 
can  readily  superintend  classes  of  about  twenty  at  a  time,  and  is 
free  to  pass  continually  from  one  to  another,  answering  questions 
and  seeing  that  no  mistakes  are  made.  He  can  also  select  such  ex- 
periments as  are  suited  to  the  requirements  or  ability  of  each  stu- 
dent, the  order  in  which  they  are  performed  being  of  little  impor- 
tance, as  the  class  is  supposed  to  have  previously  attained  a 
moderate  familiarity  with  the  general  principles  of  physics.  More- 
over, the  apparatus  never  being  moved,  the  danger  of  injury  or 
breakage  is  thus  greatly  lessened  and  much  time  is  saved.  To 
avoid  delay,  the  number  of  experiments  ready  at  any  time  should 
be  greater  than  that  of  the  students,  and  the  easier  ones  should  be 
gradually  replaced  by  those  of  greater  difficulty. 

Among  these  experiments  several  novelties,  here  published  for 
the  first  time,  have  been  introduced.  For  instance,  the  apparatus 
for  ruling  scales,  p.  59,  the  photometers,  pp.  132  and  134,  and  the 
polarimeter,  p.  221.  It  is  also  believed  that  the  directions  for 
weighing,  p.  47,  and  the  adjustments  for  the  optical  circle,  p.  142,  if 
not  new,  at  least  present  the  subject  in  a  more  concise  and  practi- 
cal form  than  that  commonly  given.  In  fact  it  has  been  the 
object  throughout  to  give  definite  directions,  so  far  as  possible,  as 
if  addressing  the  student  in  person.  English  weights  and  measures 
are  occasionally  used  as  well  as  French  to  familiarize  the  student 
with  both  systems,  as  in  many  of  the  practical  applications  of  phys- 
ics the  general  prevalence  of  the  foot  and  pound  as  units  seems 


Vlll  PREFACE. 

to  render  premature  the  exclusive  introduction  of  the  metric  sys- 
tem. The  second  volume  of  this  work,  including  Heat,  Electric- 
ity, a  list  of  books  of  reference,  and  other  matters  of  general 
interest  to  the  physicist,  will  be  issued  at  as  early  a  date  as  pos- 
sible. 

It  is  difficult  to  give  credit  for  all  the  aid  rendered  in  preparing 
this  work,  as  the  author  has  for  years  made  it  a  practice  to  collect 
for  it  information  from  all  available  sources.  He  is  much  indebted 
to  Mr.  Alvan  Clark  for  the  method  of  testing  telescope  lenses,  and 
to  Prof.  F.  E.  Stimpson  for  advice  and  aid  on  photometry  and 
other  matters.  The  course  in  photography  is  essentially  that 
given  by  Mr.  Whipple  to  the  students  at  the  Institute.  His 
especial  thanks  are  due  to  his  friend  Prof.  Cross,  whose  careful 
examination  of  the  proof  sheets,  and  whose  excellent  judgment 
has  been  of  great  assistance.  Finally,  if  this  volume,  notwith- 
standing its  shortcomings,  aids  in  any  way  those  engaged  in 
physical  investigations,  either  the  student  in  the  laboratory  or  the 
amateur  experimenter,  the  object  of  the  author  will  have  been 
accomplished. 

E.  C.  P. 

April  29^,  1873. 


INDEX, 


GENERAL  METHODS  OF  PHYSICAL  INVESTIGATION. 

ANALYTICAL  METHOD.     .........        3 

Mean,  3.     Probable  Error,  3.     Weights,  4.     Probable  Error  of 
Two  or  More  Variables,  4.     Peirce's  Criterion,  6.     Differences,  6. 
Interpolation,  7.      Inverse  Interpolation,  8.     Numerical  Computa- 
tion, 9.     Significant  Figures,  10.     Successive  Approximations,  10. 
GRAPHICAL  METHOD         .........      11 

Interpolation,  12.  Residual  Curves,  12.  Maxima  and  Minima,  IS. 
Points  of  Inflexion,  13.  Asymptotes,  14.  Curves  of  Error,  14. 
TAree  Variables,  14. 

PHYSICAL  MEASUREMENTS 16 

Time,  16.  Weight,  19.  Length,  19.  ^Ireas,  22.  Volumes,  22. 
Angles,  23.  Curvature,  25. 

GENERAL  EXPERIMENTS. 

1.  ESTIMATION  OF  TENTHS 27 

2.  VERNIERS 28 

3.  INSERTION  OF  CROSS-HAIRS ' .  29 

4.  SUSPENSION  BY  SILK  FIBRES 31 

5.  TEMPERATURE  CURVE .31 

6.  TESTING  THERMOMETERS 32 

7.  ECCENTRICITY  OF  GRADUATED  CIRCLES       ....  33 

8.  CONTOUR  LINES 34 

9.  CLEANING  MERCURY       ........  35 

10.  CALIBRATION  BY  MERCURY 37 

(ix) 


X  INDEX. 

11.  CALIBRATION  BY  WATER        . 39 

12.  CATHETOMETER 39 

13.  HOOK  GAUGE 41 

14.  SPHEROMETER •  .        .42 

15.  ESTIMATION  OF  TENTHS  OF  A  SECOND         ....  44 

16.  RATING  CHRONOMETERS 44 

17.  MAKING  WEIGHTS 46 

Proper  Method  of  Weighing   .         .         .         .         .         .         .47 

18.  DECANTING  GASES 50 

Reduction  of  Gases  to  Standard  Temperature  and  Pressure    .  5 1 

19.  STANDARDS  OF  VOLUME 52 

20.  READING  MICROSCOPES 55 

21.  DIVIDING  ENGINE 56 

22.  RULING  SCALES 59 

MECHANICS  OF  SOLIDS. 

23.  COMPOSITION  OF  FORCES 62 

24.  MOMENTS 63 

25.  PARALLEL  FORCES 64 

26.  CENTRE  OF  GRAVITY 66 

27.  CATENARY .  67 

28.  CRANK  MOTION .68 

29.  HOOK'S  UNIVERSAL  JOINT 69 

30.  COEFFICIENT  OF  FRICTION 70 

31.  ANGLE  OF  FRICTION 71 

32.  BREAKING  WEIGHT         .        .......  72 

33.  LAWS  OF  TENSION 73 

34.  CHANGE  OF  VOLUME  BY  TENSION 75 

35.  DEFLECTION  OF  BEAMS.    1 77 

36.  DEFLECTION  OF  BEAMS.     H. 79 

37.  TRUSSES 80 

38.  LAWS  OF  TORSION 82 

39.  FALLING  BODIES 84 

40.  METRONOME  PENDULUM 85 

41.  BORDA'S  PENDULUM 85 

42.  TORSION  PENDULUM 87 

MECHANICS   OF  LIQUIDS  AND   GASES. 

43.  PRINCIPLES  OF  ARCHIMEDES 89 

44.  RELATIONS  OF  WEIGHTS  AND  MEASURES  90 


INDEX.  Xi 

45.  HYDROMETERS 91 

46.  SPECIFIC  GRAVITY  BOTTLE 92 

47.  HYDROSTATIC  BALANCE          .         .        .        .        .        .        .93 

48.  EFFLUX  OF  LIQUIDS 94 

49.  JETS  OF  WATER 97 

50.  RESISTANCE  OF  PIPES 98 

51.  FLOW  OF  LIQUIDS  THROUGH  SMALL  ORIFICES    .        .        .99 

52.  CAPILLARITY 100 

53.  PLATEAU'S  EXPERIMENT 101 

54.  PNEUMATICS •••'••..  103 

55.  MARIOTTE'S  LAW 107 

56.  GAS-HOLDER 109 

57.  GAS-METERS Ill 

58.  BAROMETER 114 

Measurement  of  Heights  by  the  Barometer       .        .        .        .116 

59.  BUNSEN  PUMP 118 

60.  AIR-METER 120 

SOUND. 

61.  SIRENE 122 

62.  KUNDT'S  EXPERIMENT    .         .        .        .        .        .        .        .123 

63.  MELDE'S  EXPERIMENT 124 

64.  ACOUSTIC  CURVES  . 125 

65.  LISSAJOUS'  EXPERIMENT .128 

66.  CHLADNI'S  EXPERIMENT 130 

LIGHT. 

67.  PHOTOMETER  FOR  ABSORPTION      .        .        .        .                .  132 

68.  DAYLIGHT  PHOTOMETER 134 

69.  BUNSEN  PHOTOMETER 135 

70.  LAW  OF  REFLECTION 138 

71.  ANGLES  OF  CRYSTALS 139 

72.  ANGLE  OF  PRISMS 141 

73.  LAW  OF  REFRACTION.     1 145 

74.  LAW  OF  REFRACTION.    II 146 

75.  INDEX  OF  REFRACTION  .        .        .        .        .        .        .        .  147 

76.  CHEMICAL  SPECTROSCOPE 148 

77.  SOLAR  SPECTROSCOPE     ........  151 

78.  LAW  OF  LENSES 155 

79.  MICROSCOPE 156 

80.  PREPARATION  OF  OBJECTS     .        .        .        •        .        .        .167 


Xll  INDEX. 

81.  MOUNTING  OBJECTS        .        .        .        .        •        .        .        .170 

82.  Foci  AND  APERTURE  OF  OBJECTIVES 173 

83.  TESTING  PLANE  SURFACES 175 

84.  TESTING  TELESCOPES 178 

85.  PHOTOGRAPHY  I.     GLASS  NEGATIVES 181 

86.  PHOTOGRAPHY  II.     PAPER  POSITIVES 187 

87.  TESTING  THE  EYE 191 

88.  OPTHALMOSCOPE     .        .        .        .        .        .        .        .        .196 

89.  INTERFERENCE  OF  LIGHT 199 

90.  DIFFRACTION 202 

91.  WAVE  LENGTHS .205 

92.  POLARIZED  LIGHT .        .        .     208 

93.  POLARISCOPE 217 

94.  SACCHARIMETER      .        .        .        ....        .        .        .     222 


GENERAL  METHODS 


OP 


PHYSICAL  INVESTIGATION. 


THE  object  of  all  Physical  Investigation  is  to  determine  the 
effects  of  certain  natural  forces,  such  as  gravity,  cohesion,  heat, 
light  and  electricity.  For  this  purpose  we  subject  various  bodies 
to  the  action  of  these  forces,  and  note  under  what  circumstances 
the  desired  effect  is  produced  ;  this  is  called  an  experiment.  In- 
vestigations may  be  of  several  kinds.  First,  we  may  simply  wish 
to  know  whether  a  certain  effect  can  be  produced,  and  if  so,  what 
are  the  necessary  conditions.  To  take  a  familiar  example,  we  find 
that  water  when  heated  boils,  and  that  this  result  is  attained 
whether  the  heat  is  caused  by  burning  coal,  wood  or  gas,  or  by 
concentrating  the  sun's  rays ;  also  whether  the  water  is  contained 
in  a  vessel  of  metal  or  glass,  and  finally  that  the  same  effect 
may  be  produced  with  almost  all  other  liquids.  Such  work  is 
called  Qualitative,  since  no  measurements  are  needed,  but  only  to 
determine  the  quality  or  kind  of  conditions  necessary  for  its  fulfil- 
ment. Secondly,  we  may  wish  to  know  the  magnitude  of  the  force 
required,  or  the  temperature  necessary  to  produce  ebullition. 
This  we  should  find  to  be  about  100°  C.  or  212°  F.,  but  varying 
slightly  with  the  nature  of  the  vessel  and  the  pressure  of  the  air. 
Thirdly,  we  often  find  two  quantities  so  related  that  any  change 
in  one  produces  a  corresponding  change  in  the  other,  and  we  may 
wish  to  find  the  law  by  which  we  can  compute  the  second,  having 
given  any  value  of  the  first.  Thus  by  changing  the  pressure  to 
which  the  water  is  subjected,  we  may  alter  the  temperature  of 
boiling,  and  to  determine  the  law  by  which  these  two  quantities 
are  connected,  hundreds  of  experiments  have  been  made  by  physi- 


2  ERRORS. 

cists  in  all  parts  of  the  world.  The  last  two  classes  of  experi- 
ments are  called  Quantitative,  since  accurate  measurements  must 
be  made  of  the  quantity  or  magnitude  of  the  forces  involved. 
Most  of  the  following  experiments  are  of  this  nature,  since  they 
require  more  skill  in  their  performance,  and  we  can  test  with  more, 
certainty  how  accurately  they  have  been  done.  Having  obtained 
a  number  of  measurements,  we  next  proceed  to  discuss  them  by 
the  aid  of  the  mathematical  principles  described  below,  and  finally 
to  draw  our  conclusions  from  them.  It  is  by  this  method  that  the 
whole  science  of  Physics  has  been  built  up  step  by  step. 

Errors.  In  comparing  a  number  of  measurements  of  the  same 
quantity,  we  always  find  that  they  differ  slightly  from  one  another, 
however  carefully  they  may  be  made,  owing  to  the  imperfection  of 
all  human  instruments,  and  of  our  own  senses.  These  deviations 
or  errors  must  not  be  confounded  with  mistakes,  or  observations 
where  a  number  is  recorded  incorrectly,  or  the  experiment  improp- 
erly performed;  such  results  must  be  entirely  rejected,  and  not 
taken  into  consideration  in  drawing  our  conclusions. 

If  we  knew  the  true  value,  and  subtracted  it  from  each  of  our 
measurements,  the  differences  would  be  the  errors,  and  these  may 
be  divided  into  two  kinds.  We  have  first,  constant  errors,  such  as 
a  wrong  length  of  our  scale,  incorrect  rate  of  our  clock,  or  natural 
tendency  of  the  observer  to  always  estimate  certain  quantities  too 
great,  and  others  too  small.  When  we  change  our  variables 
these  errors  often  alter  also,  but  generally  according  to  some  defi- 
nite law.  When  they  alternately  increase  and  diminish  the  result 
at  regular  intervals  they  are  called  periodic  errors.  If  we  know 
their  magnitude  they  do  no  harm,  since  we  can  allow  for  them, 
and  thus  obtain  a  value  as  accurate  as  if  they  did  not  exist.  The 
second  class  of  errors  are  those  which  are  due  to  looseness  of  the 
joints  of  our  instruments,  impossibility  of  reading  very  small  dis- 
tances by  the  eye,  &c.,  which  sometimes  render  the  result  too  large, 
sometimes  too  small.  They  are  called  accidental  errors,  and  are 
unavoidable ;  they  must  be  carefully  distinguished  from  the  mis- 
takes referred  to  above. 

Analytical  and  Graphical  Methods.  There  are  two  ways  of 
discussing  the  results  of  our  experiments  mathematically.  By  the 
first,  or  Analytical  Method,  we  represent  each  quantity  by  a  letter, 


ANALYTICAL   METHOD.  3 

and  then  by  means  of  algebraic  methods  and  the  calculus  draw 
our  conclusions.  By  the  Graphical  Method  quantities  are  repre- 
sented by  lines  or  distances,  and  are  then  treated  geometrically. 

The  former  method  is  the  most  accurate,  and  would  generally 
be  the  best,  were  it  not  for  the  accidental  errors,  and  were  all 
physical  laws  represented  by  simple  equations.  The  Graphical 
Method  has,  however,  the  advantage  of  quickness,  and  of  enabling 
us  to  see  at  a  glance  the  accuracy  of  our  results. 

ANALYTICAL   METHOD. 

Mean.  Suppose  we  have  a  number  of  observations,  Al ,  A2 ,  A3 , 
A± ,  <fcc.,  differing  from  one  another  only  by  the  accidental  errors, 
and  we  wish  to  find  what  value  A  is  most  likely  to  be  correct.  If 
A  was  the  true  value,  A±  —  A,  A2  —  A,  &c.,  would  be  the  errors 
of  each  observation,  and  it  is  proved  by  the  Theory  of  Probabil- 
ities that  the  most  probable  value  of  A  is  that  which  makes  the 
sum  of  the  squares  of  the  errors  a  minimum.  Also  that  this  prop- 
erty is  possessed  by  the  arithmetical  mean.  Hence,  when  we 
have  n  such  observations,  we  take  A  =  (A1-}-A2-\-A8-\-  &c.)  -r-  w, 
or  divide  their  sum  by  n.  Thus  the  mean  of  32,  33,  31,  30,  34,  is 
160  -7-  5  =  32.  It  is  often  more  convenient  to  subtract  some  even 
number  from  all  the  observations,  and  add  it  to  the  mean  of  the  re- 
mainder; thus,  to  find  the  mean  of  1582,  1581,  1583,  1581,  1582, 
subtract  1580  from  each,  and  we  have  the  remainders  2,  1,  3,  1,  2. 
Their  mean  is  9  -f-  5  =  1.8,  which  added  to  1580  gives  1581.8. 
Where  many  numbers  are  to  be  added,  Webb's  Adder  may  be 
used  with  advantage. 

Probable  Error.  Having  by  the  method  just  given,  found  the 
most  probable  value  of  A,  we  next  wish  to  know  how  much  reli- 
ance we  may  place  on  it.  If  it  is  just  an  even  chance  that  the 
true  value  is  greater  or  less  than  A  by  E,  then  E  is  called  its 
probable  error.  To  find  this  quantity,  subtract  the  mean  from 
each  of  the  observed  values,  and  place  A1  —  A  =  el ,  A2  —  A 
=  02,  &c.  Now  the  theory  of  probabilities  shows  that  E  = 
.67-v/e!2  +  e22  +  &c.,  -r-  n,  from  which  we  can  compute  E  in  any 
special  case.  As  an  example,  suppose  we  have  measured  the 
height  of  the  barometer  twenty-five  times,  and  find  the  mean 
29.526  with  a  probable  error  of  .001  inches.  Then  it  is  an  even 


4  PROBABLE    ERROR. 

chance  that  the  true  reading  is  more  than  29.525,  and  less  than 
29.527.  Now  let  us  suppose  that  some  other  day  we  make  a  single 
reading,  and  wish  to  know  its  probable  error.  The  theory  of 
probabilities  shows  that  the  accuracy  is  proportional  to  the  square 
root  of  the  number  of  observations,  or  that  the  mean  of  four,  is 
only  twice  as  accurate  as  a  single  reading,  the  mean  of  a  hundred, 
ten  times  as  accurate  as  one.  Hence  in  our  example  we  have 
1  :  -v/25  =  .001  :  .005,  the  probable  error  of  a  single  reading. 
Substituting  in  the  formula,  we  have  the  probable  error  of  a  single 
reading,  E'  =  E  X  Jn  —  &lje?  +  e/  +  <fcc.  4-  \/n.  It  is  gene- 
rally best  to  compute  Er  as  well  as  E,  and  thus  learn  how  much 
dependence  can  be  placed  on  a  single  reading  of  our  instrument. 

Weights.  We  have  assumed  in  the  above  paragraph  that  all 
our  observations  are  subject  to  the  same  errors,  and  hence  are 
equally  reliable.  Frequently  various  methods  are  used  to  obtain 
the  same  result,  and  some  being  more  accurate  than  others  are  said 
to  have  greater  weight.  Again,  if  one  was  obtained  as  the  mean 
of  two,  and  the  second  of  three  similar  observations,  their  weights 
would  be  proportional  to  these  numbers,  and  the  simplest  way  to 
allow  for  the  weights  of  observations  is  to  assume  that  each  is 
duplicated  a  number  of  times  proportional  to  its  weight.  From 
this  statement  it  evidently  follows  that  instead  of  the  mean  of  a 
series  of  measurements,  we  should  multiply  each  by  its  weight, 
and  divide  by  the  sum  of  the  weights.  Calling  A1  ,  A2  ,  &c.,  the 
measurements,  and  tol5  w2,  &c.,  their  weights,  the  best  value  to  use 
will  be  A  =  (A1wi  -{-  Azw2-\-  &c.)  -f-  (w±  +  w2  +  <fcc.).  We  may 
always  compute  the  weight  of  a  series  of  n  observations,  if  we 
know  the  errors  el  ,  e2  ,  &c.,  using  the  formula  w  =  n  4-  2(e^  +  e22  + 
e*  -f-  &c.).  Substituting  this  value  in  the  equation  for  probable  error, 
we  deduce  E  =  All  4-  +/nw  if  all  the  observations  have  the 


same  weight,  or  -E'—  .477  -f-  ^/w^  +  w2  +  &c.,  if  their  weights  are 

Wl  ,  W2  ,  &C. 

Probable  Error  of  Two  or  More  Variables.  Suppose  we  have 
a  number  of  observations  of  several  quantities,  aj,  y,  2,  and  know 
that  they  are  so  connected  that  we  shall  always  have  0  =  1  -{-  ax 
+  by  +  cz.  If  the  first  term  of  the  equation  does  not  equal  1,  we 
may  make  it  so,  by  dividing  each  term  by  it.  Call  the  various 
values  x  assumes  #',  a",  #'",  those  of  y,  y',  y",  y"',  and  those 


TWO    OR    MORE    VARIABLES.  5 

of  2,  2',  2",  2"',  and  so  on  for  any  other  variables  which  may  enter. 
If  we  have  more  observations  than  variables,  it  will  not  in  general 
be  possible  to  find  any  values  of  a,  b  and  c  which  will  satisfy  them 
all,  but  we  shall  always  find  the  left  hand  side  of  our  equation 
instead  of  being  zero  will  become  some  small  quantity,  e'^  e"^  e"\  so 
that  we  shall  have :  — 

e'    =  l+axf    +V    +  cz', 

e"  =  1  +  ax"  +  ly"  +  cz", 

e'"  =1  +  axf"  +  by"'  +  cz"', 

and  so  on,  one  equation  corresponding  to  each  observation.  These 
are  called  equations  of  condition.  Now  we  wish  to  know  what 
are  the  most  probable  values  of  a,  b  and  c,  that  is,  those  which  will 
make  the  errors  er,  e",  e'",  as  small  as  possible.  As  before,  we  must 
have  the  sum  of  the  squares  of  the  errors  a  minimum.  We  there- 
fore square  each  equation  of  condition,  and  take  their  sum ;  differ- 
entiate this  with  regard  to  a,  b  and  c,  successively,  and  place  each 
differential  coefficient  equal  to  zero.  These  last  are  called  normal 
equations,  and  correspond  to  each  of  the  quantities  a,  b  and  c,  re- 
spectively. The  practical  rule  for  obtaining  the  normal  equations 
is  as  follows :  —  Multiply  each  equation  of  condition  by  its  value 
of  x  (or  coefficient  of  a),  take  their  sum  and  equate  it  to  zero. 
Thus  xr(l  -f  axf  -[-  by'  +  czr)  +  x"(l  +  ax"  +  by"  +  cz")  +  &c. 
=  0,  is  the  first  normal  equation.  Do  the  same  with  regard  to  y, 
and  each  other  variable  in  turn.  We  thus  obtain  as  many  equations 
as  there  are  quantities  a,  b  and  c  to  be  determined.  Solving  them 
with  regard  to  these  last  quantities,  and  substituting  in  the  original 
formula  0  =  1  +  ax  +  by  -f-  cz,  we  have  the  desired  equation. 

As  an  example,  suppose  we  have  the  three  points,  Fig.  1,  whose 
coordinates  are  xr  =  l,y'  =  1,  x"  =  2,  y"  =  2,  of"  =  3,  yr"  =  4, 
and  we  wish  to  pass  a  straight  line  as  nearly  as  possible  through 
them  all.     We  have  for  our   equations  of  condi- 
tions :   0  =  1  +  a  +  5,  0  =  1  +  2a  +  25,  0  = 
1  +  3a  +  4&.     Applying  our  rule,  we  multiply 
the  first  equation  by  1,  the  second  by  2,  and  the 
third  by  3,  the  three  values  of  a,  and  take  their 
sum,  which  gives  l  +  #~{~#-|-2-|-4:a-|-4£_|_3     ^ 

+  9a  +  l'2b  —  6  +  14a  +  17b  =  0.     For  our  sec-  Fig. 

ond  normal  equation  we  multiply  by  1,  2  and  4, 


6  PEIRCE'S  CRITERION. 

respectively,  and  obtain  in  the  same  way  7  +  17a  +  215  =  0. 
Solving,  we  find  a  —  — 1.4,  b  =  .8,  and  substituting  in  our  original 
equation  0  =  1  +  ax  -j-  by,  we  bave  0  =  1  —  \Ax  +  .8y,  or  y  = 
1.75aj  —  1.25.  Constructing  tbe  line  tbus  found,  we  obtain  MN, 
Fig.  1,  which  will  be  seen  to  agree  very  well  with  our  original 
conditions. 

For  a  fuller  description  of  the  various  applications  of  the  Theory 
of  Probabilities  to  the  discussion  of  observations,  the  reader  is  re- 
ferred to  the  following  works.  Methode  des  Moindres  Carrees  par 
Ch.  Fr.  Gauss,  trad,  par  J.  Bertrand,  Paris,  1855,  Watson's 
Astronomy,  360,  Chauvenefs  Astronomy,  II,  500,  and  Todhunt- 
er's  History  of  the  Theory  of  Probabilities.  A  good  brief  de- 
scription is  given  in  Darned  and  Peck's  Math.  Diet.,  454,  536  arid 
590,  also  in  Mayer's  Lecture  Notes  on  Physics,  29. 

Peirce's  Criterion.  It  has  already  been  stated  that  all  observa- 
tions affected  by  errors  not  accidental,  or  mistakes,  should  be  at 
once  rejected.  But  it  is  generally  difficult  to  detect  them,  and 
hence  various  Criteria  have  been  suggested  to  enable  us  to  decide 
whether  to  reject  an  observation  which  appears  to  differ  consid- 
erably from  the  rest.  One  of  the  best  known  of  these  is  Peirce's 
Criterion,  which  may  be  defined  as  follows :  —  The  proposed  ob- 
servations should  be  rejected  when  the  probability  of  the  system 
of  errors  obtained  by  retaining  them  is  less  than  that  of  the  system 
of  errors  obtained  by  their  rejection,  multiplied  by  the  probability 
of  making  so  many  and  no  more  abnormal  observations.  Or,  to 
put  it  in  a  simpler  but  less  accurate  form,  reject  any  observations 
which  increase  the  probable  error,  allowing  for  the  chances  of 
making  so  many  and  no  more  erroneous  measurements.  Without 
this  last  clause  we  might  reject  all  but  one,  when  the  probable 
error  by  the  formula  would  become  zero.  See  Gould's  Astron. 
Journ.,  1852,  II,  161;  IV,  81,  137,  145. 

Another  criterion  has  been  proposed  by  Chauvenet,  which, 
though  less  accurate  than  the  above,  is  much  more  easily  applied. 
It  is  fully  described  in  Watson's  Astronomy,  410. 

Differences.  To  determine  the  law  by  which  a  change  in  any 
quantity  A  alters  a  second  quantity  B,  we  frequently  measure  B 
wnen  A  is  allowed  to  alter  continually  by  equal  amounts.  Thus 
in  the  example  of  the  boiling  of  water,  we  measure  the  pressure 


INTERPOLATION. 


A.        B. 


D'. 


D".       D'". 


10 


cprresponding  to  temperatures  of  0°,  10°,  20°,  30°,  &c.  "Writing 
these  numbers  in  a  table,  by  placing  the  various  values  of  A  in 
the  first  column,  those  of  B  in 
the  second,  we  form  a  third 
column,  in  which  each  term  is 
found  by  subtracting  the  value 
of  B  from  that  preceding  it; 
the  remainders  are  called  the 
first  differences  Dr.  In  the 
same  way  we  obtain  the  sec- 
ond differences  U"t  by  sub- 
tracting each  first  difference 
from  that  which  follows  it,  and  so  on. 

Interpolation.  One  of  the  most  common  applications  of  differ- 
ences is  to  determine  the  value  of  B  for  any  intermediate  value 
of  A.  This  is  done  by  the  formula, 


0°       4.6 

+  4.6 
9.2  -f  3.6 

-f-  8.2  -j-  2.3 

20°     17.4  4-  5.9 

4-  14.1  -f-  3.4 

30°     31.5  -f  9.3 

+  23.4 
40°     54.9 


A.      B. 


in  which  Bm  is  the  measurement  next  preceding 

./>„/",  the  1st,  2d,  3d,  differences,  and  n  a  fraction  equal  to  (A  — 

^4.m)  -f-  (Am+1  —  Am),  in  which  A,  Am  correspond  to  B,  -Bm,  and 

Am+l  is  the  next  term  of  the 
series  to  Am.  The  use  of  this 
formula  is  best  shown  by  an 
example.  Suppose,  from  the 
accompanying  table,  we  wish 
to  find  the  value  of  B  corre- 
sponding to  A  =  12.5.  We 


have  Bm  =  1728,  Dm'  =  469, 
2>m"  =  78,  Dm'"  =G,Am  = 
12,  Am  +  l  =  13,  A  =  12.5 
and  n  =  (12.5  —  12)  -f-  (13 
—  12)  =  .5.  Hence, 

B  =  1728  +  .5(469)  +  ^=^(78)  + 


10  1000 

-f  331 

11  1331         4-  66 

-f  397        -f  6 

12  1728        +72       0 

'  4-469        4-6 

13  2197        4-78       0 

4-547      '4-6 

14  2744        -|-84       0 

4-631        4-6 

15  3375        4-  90 

4-  721 

16  4096 


+  0, 


B  =  1728  4-  234.5  —  9.75  4-  .375  +  0  =  1953.125. 
In  this  particular  case  B  is  always  the  cube  of  A,  and  it  may  be 


$  INVERSE   INTERPOLATION. 

seen  that  our  formula  gives  an  exact  result.    The  reason  is  that 
the  4th,  and  all  following  differences,  equal  zero. 

Inverse  Interpolation.  Next  suppose  that  in  the  above  example 
we  desired  the  value  of  A  for  some  given  value  of  .#,  as  B'\  that 
is,  in  the  equation, 


m 

1.  12  1.  2*.  o 


+  &c. 


we  wish  to  find  n.  Evidently  it  is  impossible  to  determine  this 
exactly,  but  an  'approximate  value  may  be  found  by  the  method 
of  successive  corrections.  Neglect  all  terms  after  the  third,  and 
deduce  n  from  the  equation, 


which  is  a  simple  quadratic  equation.     Substitute  this  value  of  n 
in  the  terms  we  have  neglected,  and  call  the  result  -ZVJ  then 


from  which  again  we  may  deduce  a  more  accurate  valne  of  n. 
This  again  gives  a  new  value  of  JVJ  and  by  continuing  this  process 
we  finally  deduce  n  with  any  required  degree  of  accuracy. 

It  is  sometimes  more  convenient  to  neglect  the  third  term,  and 
deduce  n  from  the  equation  IB'  =  JBm  +  n^mi  which  saves  solv- 
ing a  quadratic  equation,  but  requires  more  approximations.  The 
values  of  n(n  —  1)  -f-1.  2,  n(n  —  1)  (n  —  2)  -f-  1.  2.  3,  &c.,  may 
be  more  readily  obtained  from  Interpolation  tables  than  by  compu- 
tation. A  good  explanation  of  this  subject  is  given  in  the  Assu- 
rance Magazine,  XI,  61,  XI,  301,  and  XII,  136,  by  Woolhouse. 

When  the  terms  are  not  equidistant  the  method  of  interpolation 
by  differences  cannot  be  applied.  In  this  case,  if  we  wish  to  find 
values  of  IB  corresponding  to  known  values  of  A,  we  assume  the 
equation,  IB  =  a  +  bA  +  cAz  -f-  dA3  +  &c.,  and  see  what  values 
of  a,  5,  c,  &c.,  will  best  satisfy  these  equations.  If  we  have  a 
great  many  corresponding  values  of  A  and  j5,  the  method  of  least 
squares  should  be  applied.  In  general,  however,  it  is  much  more 
convenient  to  solve  this  problem  by  the  Graphical  Method  de- 


NUMERICAL    COMPUTATION. 


9 


scribed  below.     See  Cauchy's  Calculus,  I,  513,  and  an  article  in 
the  Connaissance  des  Temps,  for  1852,  by  Villarceau. 

Numerical  Computation.  Where  much  arithmetical  work  is 
necessary  to  reduce  a  series  of  observations,  a  great  saving  of  time 
is  effected  by  making  the  computation  in  a  systematic  form.  In 
general,  measurements  of  the  same  quantity  should  be  written  in  a 
column,  one  below  the  other,  instead  of  on  the  same  line,  and 
plenty  of  room  should  always  be  allowed  on  the  paper.  When 
the  same  computations  must  be  made  for  several  values  of  one  of 
the  variables,  instead  of  completing  one  before  beginning  the 
next,  it  is  better  to  carry  all  on  together,  as  in  the  following 
example.  Suppose,  as  in  the  experiment  of  the  Universal  Joint, 
we  wish  to  compute  the  values  of  b  in  the  formula,  tan  b  =  cos  A. 
ta;n  a,  in  which  A  =  45°,  and  a  in  turn  5°,  10°,  15°,  &c.  Con- 
struct a  table  thus :  — 


15° 


25° 


30° 


log  tan  a 

8.94195 

9.24632 

9.42805 

9.56107 

9.66867 

9.76144 

log  cos  A 

9.84948 

9.84948 

9.84948 

9.84948 

9.84948 

9.84948 

log  tan  b 

8.79143 

9.09580 

9.27753 

9.41055 

9.41815 

9.61092 

b 

3°  32' 

7°  6' 

10°  44' 

14°  26' 

14°  41' 

15' 

In  the  first  line  write  the  various  values  of  a,  in  the  second  the 
corresponding  values  of  its  log  tan,  and  so  on  throughout  the  com- 
putation. An  error  is  purposely  committed  in  the  above  table  to 
show  how  easily  it  may  be  detected.  It  will  be  noticed  that 
the  values  of  b  increase  pretty  regularly,  except  that  when  a  —  25°, 
and  that  this  is  but  little  greater  than  that  corresponding  to  a  = 
20°.  Following  the  column  up  we  find  that  the  same  is  the  case  for 
log  tan  b  but  not  for  log  tan  a,  hence  the  error  is  between  the  two. 
In  fact,  in  the  addition  of  the  logarithms  we  took  6  and  4  equal 
to  10,  and  omitted  to  carry  the  1 ;  log  tan  b  then  really  equals 
9.51815,  and  b  =  18°  15'.  If  the  error  is  not  found  at  once  this 
value  of  b  should  be  recomputed.  Besides  these  advantages,  this 
method  is  much  quicker  and  less  laborious.  When  we  have  to 
multiply,  or  divide  by,  the  same  number  A  a  great  many  times,  it 
is  often  shorter  to  obtain  at  once  1^4,  2.4,  3.4,  4^4,  &c.,  and  use 
these  numbers  instead  of  making  the  multiplication  each  time. 
This  is  useful  in  reducing  metres  to  inches,  &c.  There  are  many 


10  SIGNIFICANT   FIGURES. 

other  arithmetical  devices,  but  their  consideration  would  lead  us 
too  far  from  our  subject. 

Significant  Figures.  One  of  the  most  common  mistakes  in 
reducing  observations  is  to  retain  more  decimal  places  than  the 
experiment  warrants.  For  instance,  suppose  we  are  measuring  a 
distance  with  a  scale  of  millimetres,  and  dividing  them  into  tenths 
by  the  eye,  we  find  it  32.7  mm.  Now  to  reduce  it  to  inches  we 
have  1  metre  =  39.37  in.,  hence  32.7  mm.  =  1.287399.  But  it  is 
absurd  to  retain  the  last  three  figures,  since  in  our  original  meas- 
urement, as  we  only  read  to  tenths  of  a  millimetre,  we  are  always 
liable  to  an  error  of  one  half  this  amount,  or  .002  of  an  inch. 
Then  we  merely  know  that  our  distance  lies  between  1.2894  and 
1.2854  inches,  showing  that  even  the  thousandths  are  doubtful.  It 
is  worse  than  useless  to  retain  more  figures,  since  they  might  mis- 
lead a  reader  by  making  him  think  greater  accuracy  of  measure- 
ment had  been  attained. 

If  we  are  sure  that  our  errors  do  not  exceed  one  per  cent,  of  the 
quantity  measured,  we  say  that  we  have  two  significant  figures,  if 
one  tenth  of  a  per  cent,  three,  if  one  hundredth,  four.  Thus  in  the 
example  given  above,  if  we  are  sure  the  distance  is  nearer  32.7 
than  32.8  or  32.6,  we  have  three  significant  figures,  and  it  would 
be  the  same  if  the  number  was  327,000,  or  .00327.  In  general, 
count  the  figures,  after  cutting  off  the  zeros  at  either  end,  unless 
they  are  obtained  by  the  measurement ;  thus  300,000  has  three  sig- 
nificant figures  if  we  know  that  it  is  more  correct  than  301,000  or 
299,000.  In  reducing  results  we  should  never  retain  but  one  more 
significant  figure  than  has  been  obtained  in  the  first  measurement, 
and  must  remember  that  the  last  of  these  figures  is  sometimes 
liable  to  an  error  of  several  units. 

Successive  Approximations.  This  method  is  also  known  as 
that  of  trial  and  error.  It  consists  in  assuming  an  approximate 
value  of  the  magnitude  to  be  constructed,  measuring  the  error, 
correcting  by  this  amount  as  nearly  as  we  can,  measuring  again, 
and  so  on,  until  the  error  is  too  small  •  to  do  any  harm.  As  an 
example,  suppose  we  wish  to  cut  a  plate  of  brass  so  that  its  weight 
shall  be  precisely  100  grammes.  We  first  cut  a  piece  somewhat 
too  large,  weigh  it  and  measure  its  area.  If  its  thickness  and 
density  were  perfectly  uniform  we  could  at  once,  by  the  rule  of 


GRAPHICAL   METHOD.  11 

three,  determine  the  exact  amount  to  be  cut  off.  As,  however,  it 
will  not  do  to  make  it  too  light,  we  cut  off  a  somewhat  less  quan- 
tity and  weigh  again ;  by  a  few  repetitions  of  this  process  we  may 
reduce  the  error  to  a  very  small  amount. 

This  method  is  sometimes  the  only  one  available,  but  it  should 
not  be  too  generally  used,  as  it  encourages  guessing  at  results,  and 
tends  to  destroy  habits  of  accuracy. 

GRAPHICAL   METHOD. 

Suppose  that  we  have  any  two  quantities,  x  and  y,  so  connected 
that  a  change  in  one  alters  the  other.  Then  we  may  construct  a 
curve,  in  which  abscissas  represent  various  values  of  £c,  and  ordi- 
nates  the  corresponding  values  of  y.  Thus  suppose  we  know  that 
y  is  always  equal  to  twice  x.  Take  a  piece  of  paper  divided  into 
squares  by  equidistant  vertical  and  horizontal  lines.  Select  one  of 
each  of  these  lines  to  start  from.  The  vertical  one  is  called  the 
axis  of  Y,  the  other  the  axis  of  X,  and  their  intersection,  the  ori- 
gin. Make  x  =  1,  y  will  equal  2,  since  it  is  double  x\  now  con- 
struct a  point  distant  1  space  from  the  origin  horizontally,  and  2 
vertically.  Make  x  =  2,  y  =  4,  and  we  have  a  second  point; 
x  —  — 1,  gives  y  =  — 2,  &c.,  and  x  —  0,  gives  y  =  0.  Connecting 
these  points  we  get  a  straight  line  passing  through  the  origin,  as  is 
evident  by  analytical  geometry  from  its  equation,  y  =  2x.  Again, 
let  y  always  equal  the  square  of  cc,  and  we  have  the  corresponding 
values  x  =  Q,  y  =  0 ;  «j  =  1,  y  =  1 ;  x  =  — 1,  y  =  1 ;  x  =  2, 
y  =  4 ;  connecting  all  the  points  thus  found  we  obtain  a  par- 
abola with  its  apex  at  the  origin,  and  tangent  to  the  axis  of  JX7 
As  another  example,  suppose  we  have  made  a  series  of  experiments 
on  the  volumes  of  a  given  amount  of  air  corresponding  to  different 
pressures.  Construct  points  making  horizontal  distances  volumes, 
and  vertical  distances  pressures.  It  will  be  found  that  a  smooth 
curve  drawn  through  these  points  approaches  closely  to  an  equi- 
lateral hyperbola  with  the  two  axes  as  asymptotes.  Now  this 
curve  has  the  equation  xy  =  a,  or  y  =  a  -f-  ic,  that  is,  the  volume 
is  inversely  proportional  to  the  pressure,  which  is  Mariotte's  law. 
Owing  to  the  accidental  errors  the  points  will  not  all  lie  on  the 
curve,  but  some  will  be  above  it  and  others  below,  and  this  will  be 
true  however  many  points  may  be  observed. 


12  INTERPOLATION. 

In  general,  then,  after  observing  any  two  quantities,  A  and  B, 
construct  points  such  that  their  ordinates  and  abscissas  shall  be 
these  quantities  respectively.  Draw  a  smooth  curve  as  nearly  as 
possible  through  them,  and  then  see  if  it  coincides  with  any  com- 
mon curve,  or  if  its  form  can  be  defined  in  any  simple  way.  To 
acquire  practice  in  using  the  Graphical  Method  it  is  well  to  con- 
struct a  number  of  curves  representing  familiar  phenomena,  as 
the  variation  in  the  U.  S.  debt  during  the  late  war,  the  strength 
of  horses  moving  at  different  rates,  and  the  alterations  of  the  ther- 
mometer during  the  day  or  year.  It  is  by  no  means  necessary  that 
the  same  scale  should  be  used  for  vertical,  as  for  horizontal  dis- 
tances, but  this  should  depend  on  the  size  of  paper,  making  the 
curve  as  large  as  possible.  The  greatest  accuracy  is  attained  when 
the  latter  is  about  equally  inclined  to  both  axes. 

It  is  sometimes  better  when  one  of  the  variables  is  an  angle  to 
use  polar  coordinates.  In  this  case  paper  must  be  used  with  a 
graduated  circle  printed  on  it.  The  points  are  constructed  by 
drawing  lines  from  the  centre  in  the  direction  represented  by  one 
variable,  and  measuring  off  on  them  distances  equal  to  the  other. 
For  ordinary  purposes  circles  may  easily  be  drawn,  and  divided 
with  sufficient  accuracy  by  hand.  Laying  off  the  radius  on  the 
circumference  divides  it  to  60° ;  bisecting  these  spaces  gives  30°, 
and  a  second  bisection  15°.  By  trial  these  angles  may  be  divided 
into  three  equal  parts,  which  is  generally  small  enough,  as  the  ob- 
servations are  usually  taken  at  intervals  of  5°. 

Interpolation.  All  kinds  of  interpolation  are  very  readily  per- 
formed by  the  Graphical  Method.  After  constructing  one  curve 
to  find  the  value  of  y,  for  any  given  value  of  x  as  a/,  we  have  only 
to  draw  a  line  parallel  to  the  axis  of  YJ  at  a  distance  xf,  and  note 
the  ordinate  of  the  point  where  it  meets  the  curve.  Inverse  inter- 
polation is  performed  in  the  same  manner,  and  this  method  is 
equally  applicable,  whether  the  observations  are  at  equal  intervals 
or  not.  As  by  drawing  a  smooth  curve  the  accidental  errors  are 
in  a  great  measure  corrected,  this  method  of  interpolation  is  often 
more  accurate  than  that  by  differences. 

Residual  Curves.  The  principal  objection  to  the  Graphical 
Method,  as  ordinarily  used,  is  its  inaccuracy,  as  by  it  we  can  rarely 
obtain  more  than  three  significant  figures,  although  Regnault,  by 


RESIDUAL    CURVES.  13 

using  a  large  plate  of  copper  and  a  dividing  engine  to  construct 
his  points,  attained  a  higher  degree  of  precision. 

It  will  be  found,  however,  that  in  many  of  the  most  carefully 
conducted  researches  the  fourth  figure  is  doubtful,  as  for  example? 
in  Regnault's  measurements  of  the  pressure  of  steam,  and  even  in 
Angstrom's  and  Van  der  Willingen's  determinations  of  wave- 
lengths. 

By  the  following  device  the  accuracy  of  the  Graphical  Method 
may  be  increased  almost  indefinitely.  After  constructing  our 
points,  assume  some  simple  curve  passing  nearly  through  them. 
From  its  equation  compute  the  value  of  y  for  each  observed  value 
of  a,  and  construct  points  whose  ordinates  shall  equal  the  differ- 
ence between  the  point  and  curve  on  an  enlarged  scale,  while  the 
abscissas  are  unchanged.  Thus  let  a/,  yr  be  the  observed  coordi- 
nates, and  y  =f(x),  the  assumed  curve.  Construct  a  new  point, 
whose  coordinates  are  x'  and  a  \_yr  — /(«')]>  m  which  a  equals  5, 
10,  or  100,  according  to  the  enlargement  desired. 

Do  the  same  for  all  the  other  points,  and  a  curve  drawn  through 
them  is  called  a  residual  curve.  In  this  way  the  accidental  errors 
are  greatly  enlarged,  and  any  peculiarities  in  the  form  of  the  curve 
rendered  much  more  marked.  If  the  points  still  fall  pretty  regu- 
larly, we  may  construct  a  second  residual  curve,  and  thus  keep  on 
until  the  accidental  errors  have  attained  such  a  size  that  they  may 
be  easily  observed.  To  find  the  value  of  y  corresponding  to  any 
given  value  of  a?,  as  a?n,  we  add/(a;n)  to  the  ordinate  of  the  cor- 
responding point  of  the  residual:  curve,  first  reducing  them  to  the 
same  scale.  Most  of  the  singular  points  of  a  curve  are  very 
readily  found  by  the  aid  of  a  residual  curve.  See  an  article  by  the 
author,  Journal  of  the  Franklin  Institute,  LXI,  272. 

Maxima  and  Minima.  To  find  the  highest  point  of  a  curve, 
use,  as  an  approximation,  a  straight  line  parallel  to  the  axis  of  J£J 
and  nearly  tangent  to  the  curve.  Construct  a  residual  curve, 
which  will  show  in  a  marked  manner  the  position  of  the  required 
point.  The  same  plan  is  applicable  to  any  other  maximum  or 
minimum. 

Points  of  Inflexion.  Draw  a  line  approximately  tangent  to 
the  curve  at  the  required  point.  In  the  residual  curve  the  change 
of  curvature  becomes  very  marked. 


14  ASYMPTOTES. 

Asymptotes.  Asymptotes  present  especial  difficulties  to  the 
Graphical  Method,  as  ordinarily  used.  Suppose  our  curve  asymp- 
totic to  the  axis  of  -XT;  construct  a  new  curve  with  ordinates 
unchanged,  and  abscissas  the  reciprocals  of  those  previously  used, 
that  is  equal  to  1  -j-  x.  It  will  contain  between  0  and  1  all  the 
points  in  the  original  curve  between  1  and  QO  .  It  will  always  pass 
through  the  origin,  and  unless  tangent  to  the  axis  of  X  at  this 
point  the  area  included  between  the  curve  and  its  asymptote  will 
be  infinite.  When  this  space  is  finite,  it  may  be  measured  by  con- 
structing another  curve  with  abscissas  as  before  equal  to  1  -f-  x, 
and  ordinates  equal  to  the  area  included  between  the  curve  and 
axis,  as  far  as  the  point  under  consideration.  Find  where  this 
curve  meets  the  axis  of  Y,  and  its  ordinate  gives  the  required 
area.  A  problem  in  Diffraction  is  solved  by  this  device  in  the 
Journal  of  the  Franklin  Institute,  LIX,  264. 

Curves  of  Error.  This  very  fruitful  application  of  the  Graphi- 
cal Method  is  best  explained  by  an  example.  Suppose  we  wish  to 
draw  a  tangent  to  the  curve  B'A,  Fig.  2,  at  the  point  A.  Describe 
a  circle  with  A  as  a  centre,  through  which 
pass  a  series  of  lines,  as  AB,  AD,  AE. 
Now  construct  C  by  laying  off  BC  equal 
to  AB',  the  part  of  the  curve  cut  off  by 
the  line.  We  thus  get  a  curve  CD, 
called  the  curve  of  error,  intersecting  the 
circle  at  D,  and  the  line  AD  is  the  re- 
quired tangent.  This  is  evident,  since  if 
we  made  our  construction  at  this  point  we 

should  have  no  distance  intercepted,  or  the  line  AD  touching,  but 
not  cutting,  the  curve.  A  similar  method  may  be  applied  to  a 
great  variety  of  problems,  such  as  drawing  a  tangent  parallel  to  a 
given  line,  or  through  a  point  outside  the  curve. 

Three  Variables.  The  Graphical  Method  may  also  be  applied 
where  we  have  three  connected  variables.  If  we  construct  points 
whose  coordinates  in  space  equal  these  three  variables,  a  surface  is 
generated  whose  properties  show  the  laws  by  which  they  are  con- 
nected. To  represent  this  surface  the  device  known  as  contour 
lines  may  be  used,  as  in  showing  the  irregularities  of  the  ground 
in  a  map.  First,  generate  a  surface  by  constructing  points  in  which 


CONTOUR   LINES.  15 

ordinates  and  abscissas  shall  correspond  to  two  of  the  variables,  and 
mark  near  each  in  small  letters  the  magnitude  of  the  third  varia- 
ble, which  represents  its  distance  from  the  plane  of  the  paper.  If 
now  we  pass  a  series  of  equidistant  planes  parallel  to  the  paper, 
their  intersections  with  the  surface  will  give  the  required  contour 
lines.  To  find  these  intersections,  connect  each  pair  of  adjacent 
points  by  a  straight  line,  and  mark  on  it  its  intersections  with 
the  intervening  parallel  planes.  Thus  if  two  adjacent  points  have 
elevations  of  28  and  32,  we  may  regard  the  point  of  the  surface 
midway  between  them,  as  at  the  height  30,  or  as  lying  on  the  30 
contour  line.  Construct  in  this  way  a  number  of  points  at  the 
same  height,  and  draw  a  smooth  curve  approximately  through 
them ;  do  the  same  for  other  heights,  and  we  thus  obtain  as  many 
contours  as  we  please. 

They  give  an  excellent  idea  of  the  general  form  of  the  surface, 
and  by  descriptive  geometry  it  is  easy  to  construct  sections  passing 
through  the  surface  in  any  direction.  An  easy  way  to  understand 
the  contours  on  a  map  is  to  imagine  the  country  flooded  with  wa- 
ter, when  the  contours  will  represent  the  shore  lines  when  the 
water  stands  at  different  heights.  This  method  is  constantly  used 
in  Meteorology  to  show  what  points  have  equal  temperature,  pres- 
sure, magnetic  variation,  &c.  Contour  lines  follow  certain  general 
laws  which  are  best  explained  by  regarding  them  as  shore  lines,  as 
described  above.  Thus  contour  lines  have  no  terminating  points ; 
they  must  either  be  ovals,  or  extend  to  infinity.  Two  contours 
never  touch  unless  the  surface  becomes  vertical,  nor  cross,  unless  it 
overhangs.  A  single  contour  line  cannot  lie  between  two  others, 
both  greater  or  both  smaller,  unless  we  have  a  ridge  or  gulley  per- 
fectly horizontal,  and  at  precisely  the  height  of  the  contour.  In 
general,  such  lines  should  be  drawn  either  as  a  series  of  long  ovals, 
or  as  double  throughout.  There  will  be  no  angles  in  the  contour 
lines  unless  there  are  sharp  edges  in  the  original  surface.  A  con- 
tour line  cannot  cross  itself,  forming  a  loop,  unless  the  highest 
point  between  two  valleys,  or  the  lowest  point  between  two  hills, 
is  exactly  at  the  height  of  the  contour. 

The  value  of  contour  lines  in  showing  the  relation  between  any 
three  connected  variables,  is  well  illustrated  in  a  paper  by  Prof.  J. 


16  PHYSICAL   MEASUREMENTS. 

Thomson,  Proc.  of  the  Royal  Society,  Nov.,  1871,  also  in  Nature, 
Y,  106. 

To  acquire  facility  in  using  the  Graphical  Method,  it  is  well  to 
apply  it  to  some  numerical  examples.  Thus  take  the  equation 
y  =  axs  +  &c2  —  ex  -\-  d,  assume  certain  values  of  a,  5,  c  and  d, 
and  compute  the  value  of  y  for  various  values  of  x.  We  thus  get 
a  curve  with  two  maxima  or  minima,  and  a  point  of  inflexion. 
Find  their  position  first  by  residual  curves,  and  then  by  the  calcu- 
lus, and  see  if  they  agree.  In  the  same  way  the  curve  ya?2  —  Zayx 
-}-a2y  =  #,  has  the  axis  of  X  for  an  asymptote.  Assume,  as  before, 
positive  values  of  a  and  6,  and  determine  the  area  between  the 
curve  and  asymptote,  first  by  construction  and  then  analytically. 

PHYSICAL   MEASUREMENTS. 

The  measurement  of  all  physical  constants  may  be  divided  into 
the  determination  of  time,  of  weight  and  of  distance,  the  appara- 
tus used  varying  with  the  magnitude  of  the  quantity  to  be  meas- 
ured and  the  degree  of  accuracy  required. 

Measurement  of  Time.  A  good  clock  with  a  second  hand,  and 
beating  seconds,  should  be  placed  in  the  laboratory,  where  it  can 
be  used  in  all  experiments  in  which  the  time  is  to  be  recorded. 
Watches  with  second-hands  do  not  answer  as  well,  as  they  gener- 
ally give  five  ticks  in  two  seconds,  or  some  other  ratio  which  ren- 
ders a  determination  of  the  exact  time  difficult.  The  true  time 
may  be  measured  by  a  sextant  or  transit,  as  described  in  Experi- 
ment 16.  This  should  be  done,  if  possible,  every  clear  day  by 
different  students,  and  a  curve  constructed,  in  which  abscissas 
represent  days,  and  ordinates  errors  of  the  clock,  or  its  deviations 
from  true  time.  Short  intervals  of  time  may  be  roughly  measured 
by  a  pendulum,  made  by  tying  a  stone  to  a  string,  or  better,  by  a 
tape-measure  drawn  out  to  a  fixed  mark.  We  can  thus  measure 
such  intervals  as  the  time  of  flight  of  a  rocket  or  bomb-shell,  the 
distance  of  a  cannon  or  of  lightning,  by  the  time  required  by 
sound  to  traverse  the  intervening  space,  or  the  velocity  of  waves, 
by  the  time  they  occupy  in  passing  over  a  known  distance.  After 
the  experiment  we  reduce  the  vibrations  to  seconds  by  swinging 
our  pendulum,  and  counting  the  number  of  oscillations  per  minute. 


MEASUREMENT    OF    TIME.  17 

By  graduating  the  tape  properly,  we  may  readily  construct  a  very 
serviceable  metronome. 

Where  the  greatest  accuracy  is  required,  as  in  astronomical  ob- 
servations, a  chronograph  is  used.  A  cylinder  covered  with  paper 
is  made  to  revolve  with  perfect  uniformity  once  in  a  minute.  A 
pen  passes  against  this,  and  receives  a  motion  in  the  direction  of 
the  axis  of  the  cylinder,  of  about  a  tenth  of  an  inch  a  minute, 
causing  it  to  draw  a  long  helical  line.  An  electro-magnet  also  acts 
on  the  pen,  so  that  when  the  circuit  is  made  and  broken,  the  latter 
is  drawn  sideways,  making  a  jog  in  the  line.  To  use  this  appara- 
tus a  battery  is  connected  with  the  electro-magnet,  and  the  pendu- 
lum of  the  observatory  clock  included  in  the  circuit,  so  that  every 
second,  or  more  commonly  every  alternate  second,  the  circuit  is 
made  for  an  instant  and  then  broken.  Wires  are  carried  to  the 
observer,  who  may  be  in  any  part  of  the  building,  or  even  at  a 
distance  of  many  miles,  and  whenever  he  wishes  to  mark  the  time 
of  any  event,  as  the  transit  of  a  star,  he  has  merely,  by  a  finger 
key  (such  as  is  used  in  a  telegraph  office),  to  close  the  circuit, 
when  it  is  instantly  recorded  on  the  cylinder.  When  the  observa- 
tions are  completed  the  paper  is  unrolled  from  the  cylinder,  and  is 
found  to  be  traversed  by  a  series  of  parallel  straight  lines,  Fig.  3, 
one  corresponding  to  each  minute,  with  indentations  corresponding 
to  every  two  seconds.  The  time 
may  be  taken  directly  from  it,  the  4  — 
fractions  of  a  second  being  meas- 
ured by  a  graduated  scale.  One 
great  difficulty  in  making  this  ap- 
paratus was  to  render  the  motion 

of  the  cylinder  perfectly  uniform,  as  if  driven  by  clock-work  it 
would  go  with  a  jerk  each  second.  This  is  avoided  by  a  device  known 
as  Bond's  spring  governor,  in  which  a  spring  alternately  retards 
and  accelerates  a  revolving  axle  when  it  moves  faster  or  slower 
than  the  desired  rate.  The  seconds  marks  form  a  very  delicate 
test  for  the  regularity  of  this  motion,  since  in  consecutive  minutes 
they  should  lie  precisely  in  line,  and  the  least  variation  is  very 
marked  in  the  finished  sheet.  It  is  a  very  simple  matter  by  this 
apparatus  to  measure  the  difference  in  longitude  of  two  points.  It  is 
merely  necessary  that  an  observer  should  be  placed  at  each  station, 


18  MEASUREMENT    OF    TIME. 

with  a  transit  and  finger  key,  a  telegraph  connecting  them  with 
the  chronograph.  They  watch  the  same  star  as  it  approaches  their 
meridian,  and  each  taps  on  his  finger  key  the  instant  it  crosses  the 
vertical  line  of  his  transit.  Two  marks  are  thus  made  on  the  chro- 
nograph, and  the  interval  between  them  gives  the  difference  in 
longitude.  The  advantage  of  this  method  of  taking  transits  is  not 
so  much  its  accuracy,  as  the  ease  and  rapidity  with  which  it  is  used. 
Observers  can  work  much  longer  with  it  without  fatigue,  and  can 
use  many  more  transit  wires,  thus  greatly  increasing  the  number 
of  their  observations.  It  is  called  the  American  or  telegraphic 
method,  in  distinction  from  the  old,  or  "  eye  and  ear "  method  of 
observing  transits,  where  the  fractions  of  a  second  were  estimated, 
as  described  in  Experiment  15. 

The  chronograph  is  exceedingly  convenient  in  all  physical  in- 
vestigations where  time  is  to  be  measured,  and  nothing  but  its 
expense  prevents  its  more  general  application. 

A  simple  means  of  measuring  small  intervals  of  time  with  accu- 
racy, is  to  allow  a  fine  stream,  of  mercury  to  flow  from  a  small 
orifice,  and  collect  and  weigh  the  amount  passed  during  the  time 
to  be  measured.  Comparing  this  with  the  flow  per  minute  we 
obtain  the  time.  A  less  accurate,  but  much  more  convenient, 
liquid  for  this  purpose  is  water,  using,  in  fact,  a  kind  of  clepsydra. 
Where  very  minute  intervals  of  time  are  to  be  measured  they 
are  commonly  compared  with  the  vibrations  of  a  tuning-fork  in- 
stead of  a  pendulum.  A  fine  brass  point  is  attached  to  the  fork 
which  is  kept  vibrating  by  an  electro-magnet.  If  a  plate  of  glass 
or  piece  of  paper  covered  with  lampblack,  is  drawn  rapidly  past 
the  brass  point,  a  sinuous  line  is  drawn,  the  sinuosities  denoting 
equal  intervals  of  time,  whose  magnitude  is  readily  determined 
when  we  know  the  pitch  of  the  fork.  A  second  brass  point  is 
placed  by  the  side  of  the  fork  and  depressed  from  the  beginning  to 
the  end  of  the  time  to  be  measured.  The  length  of  the  line  thus 
drawn,  compared  with  the  sinuosities,  gives  the  time  with  great 
accuracy.  Recently  a  clock  has  been  constructed,  in  which  the 
pendulum  is  replaced  by  a  reed  vibrating  one  thousand  times  a 
second.  The  clock  is  started  and  stopped,  so  that  it  is  going  only 
during  the  time  to  be  measured,  and  the  hands  record  the  number 


MEASUREMENT    OF    WEIGHT.  19 

of  vibrations  made.     The  reed  produces  a  musical  note,  and  any 
irregularity  is  at  once  detected  by  a  change  in  its  pitch. 

Measurement  of  Weight.  This  is  done  almost  exclusively  by 
the  ordinary  balance,  whose  principle  is  so  fully  explained  in  any 
good  text-book  of  Physics  that  a  detailed  description  is  unneces- 
sary here.  We  test  the  equality  in  length  of  its  arms  by  double 
weighing,  that  is,  placing  any  heavy  body  first  in  one  pan  and  then 
in  the  other,  and  seeing  if  the  same  weights  are  required  to  coun- 
terpoise it  in  each  case.  The  center  of  gravity  should  be  very 
slightly  below  the  knife-edges.  If  too  low  the  sensibility  is  dimin- 
ished, if  too  high  the  balance  will  overturn,  and  if  coincident  with 
them  the  beam,  if  inclined,  will  not  return  to  a  horizontal  position 
The  three  knife-edges  must  be  in  line,  otherwise  the  centre  ot 
gravity  will  vary  with  the  weight  in  the  scale  pans,  and  of  course 
the  friction  must  be  reduced  to  a  minimum.  A  high  degree  of 
accuracy  may  be  obtained  with  even  an  ordinary  balance  by  first 
counterpoising  the  body  to  be  weighed,  then  removing  it  and  not- 
ing what  weights  are  necessary  to  bring  the  beam  again  to  a  hori- 
zontal position.  A  spring  balance  is  sometimes  convenient  for 
rough  work,  from  the  rapidity  with  which  it  can  be  used.  It  may 
be  rendered  quite  accurate,  though  wanting  in  delicacy,  by  noting 
the  weight  required  to  bring  its  index  to  a  certain  point,  first  when 
the  body  to  be  weighed  is  on  the  scale  pan,  and  then  when  it  is 
removed. 

Measurement  of  length.  Distances  are  most  commonly  meas- 
ured by  a  scale  of  equal  parts,  that  is,  one  with  divisions  at  regular 
intervals,  as  millimetres,  tenths  of  an  inch,  &c.  This  scale  is  then 
placed  opposite  the  distance  to  be  measured,  and  the  reading  taken 
directly.  To  obtain  greater  accuracy  than  within  a  single  division, 
we  may  divide  them  into  tenths  by  the  eye,  as  in  Experiment  1. 
The  steel  scales  of  Brown  &  Sharpe  are  good  for  common  measure- 
ments, and  may  be  obtained  with  either  English  or  French  gradua- 
tion. Instead  of  dividing  into  tenths  by  the  eye,  a  vernier  is 
frequently  used.  Thus  to  read  a  millimetre  scale  to  tenths,  nine 
spaces  are  divided  into  ten  equal  parts,  each  of  which  will  be  a 
tenth  of  a  millimetre  less  than  the  divisions  of  the  scale,  as  in  Ex 
periment  2. 

One  of  the  best  devices  for  measuring  very  minute  quantities  is 


20  MEASUREMENT    OF    LENGTH. 

the  micrometer  screw.  A  divided  circle  is  attached  to  the  head  of 
a  carefully  made  screw,  so  that  a  large  motion  of  the  former  cor- 
responds to  a  very  minute  motion  of  the  latter.  Thus  if  the  pitch 
of  the  screw  is  one  millimetre,  and  the  circle  is  divided  into  one 
hundred  parts,  turning  it  completely  around  will  move  the  screw 
but  one  millimetre,  or  turning  it  through  one  division  only  one 
hundredth  of  a  millimetre.  One  of  the  best  examples  of  this  in- 
strument is  the  dividing  engine,  which  consists  of  a  long  and  very 
perfect  micrometer  screw  with  a  movable  nut.  See  Experiment  21, 
also  Jamirfs  Physics,  I,  25.  It  is  much  used  in  engraving  scales, 
but  it  has  certain  defects  which  are  unavoidable,  and  have  caused 
some  of  our  best  mechanicians  to  give  it  up.  For  example,  it  is 
impossible  to  make  a  screw  perfectly  accurate,  and  every  joint,  of 
which  there  are  several,  is  a  source  of  constantly  varying  error. 
For  these  reasons,  and  owing  to  its  expense,  the  instrument  de- 
scribed in  Experiment  22  is  for  many  purposes  preferable.  Two 
blocks  of  wood  are  drawn  forward  alternately  step  by  step,  through 
distances  regulated  by  the  play  of  a  peg  between  a  plate  of  brass 
and  the  end  of  a  screw.  As  all  joints  are  thus  avoided,  and  the 
interval  is  determined  by  the  direct  contact  of  two  pieces  of  metal, 
great  accuracy  is  attainable  by  it. 

Where  several  scales  are  to  be  made  with  the  utmost  accuracy, 
one  should  first  be  divided  as  correctly  as  possible,  and  its  errors 
carefully  studied  by  comparing  the  different  parts  with  one  an- 
other, or  with  a  standard.  It  may  then  be  copied  by  laying  it  on 
the  same  support  with  one  of  the  other  scales,  and  moving  both  so 
that  one  shall  pass  under  a  reading  microscope,  the  other  under  a 
graver.  We  may  thus  copy  any  scale  with  great  accuracy,  but  the 
process  is  very  laborious.  A  good  way  to  construct  the  first  scale 
is  by  continual  bisection  with  beam  compasses,  as  is  done  in  grad- 
uatino-  circles.  The  finest  scales  are  ruled  with  a  diamond  on 

O 

glass.  M.  Nobert  has  succeeded  in  making  them  with  divisions 
of  less  than  a  hundred  thousandth  of  an  inch.  The  intervals 
are  so  minute  that  until  within  a  few  years  no  microscope  could 
separate  the  lines.  The  method  of  making  them  is  kept  a  secret. 
Mr.  Peters,  by  a  combination  of  levers,  has  succeeded  in  reducing 
writings  or  drawings  to  less  than  one  six  thousandth  their  original 
size.  He  exhibited  some  writing  done  by  this  machine,  which 


MINUTE    MEASUREMENTS.  21 

was  so  minute  that  the  whole  Bible  might  be  written  twenty-seven 
times  in  a  square  inch.  Finally,  it  is  claimed  that  Mr.  Whitworth 
was  able  to  detect  differences  of  one  millionth  of  an  inch  with  a 
micrometer  screw  he  has  constructed. 

To  measure  very  minute  distances  a  microscope  is  often  used 
with  a  scale  inserted  in  its  eyepiece,  which  is  used  like  a  common 
rule.  The  absolute  size  of  the  divisions  must  be  determined  be- 
forehand by  measuring  with  it  a  standard  millimetre,  or  hundredth 
of  an  inch.  A  more  accurate  method,  however,  is  the  spider-line 
micrometer,  in  which  a  fine  thread  is  moved  across  the  field  of  view 
by  a  micrometer  screw,  and  small  distances  thus  measured  with 
the  greatest  precision.  By  using  two  of  these  instruments,  which 
are  then  called  reading  microscopes,  larger  distances  may  be  meas- 
ured, or  standards  of  length  compared,  as  in  Experiment  20. 

Small  distances  are  also  sometimes  measured  by  a  lever,  with 
one  arm  much  longer  than  the  other,  so  that  a  slight  motion  of  the 
latter  is  shown  on  a  greatly  magnified  scale.  Instead  of  a  long 
arm  it  is  better  to  use  a  mirror,  and  view  in  it  the  image  of  a  scale 
by  a  telescope.  An  exceedingly  small  deviation  is  thus  readily 
perceptible,  and  this  arrangement,  sometimes  known  as  Saxton's 
pyrometer,  has  been  applied  to  a  great  variety  of  uses.  Where  we 
wish  to  bring  the  lever  always  into  the  same  position  a  level  may 
be  substituted  for  the  mirror,  forming  the  instrument  called  the 
contact  level.  Small  distances  are  also  sometimes  measured  by  a 
wedge  with  very  slight  taper,  but  this  plan  is  objectionable  on 
many  accounts.  In  geodesy  all  the  measurements  are  dependent 
on  the  accurate  determination  in  the  first  place  of  a  distance  of 
five  or  ten  miles,  called  a  base  line.  Most  of  the  above  devices 
have  been  tried  on  such  lines ;  thus  the  reading  microscope  was 
used  by  Colby  in  the  Irish  survey,  the  wedge  in  Hanover,  and  by 
Bessel  in  Prussia,  the  lever  by  Struve  in  Russia,  and  the  contact 
level  is  now  in  use  on  our  Coast  Survey.  The  principle  in  all  is  to 
use  two  long  bars  alternately,  which  are  either  brought  in  contact, 
or  the  distance  between  their  ends  measured  each  time  they  are 
laid  down. 

Many  other  physical  constants  are  really  determined  by  a  meas- 
ure of  length.  Thus  temperatures  are  determined  by  a  scale  of 
equal  parts  in  the  thermometer,  and  here  sufficient  accuracy  is  ob- 


22  AREAS    AND    VOLUMES. 

tained  by  reading  with  the  unaided  eye.  Pressures  of  air  and 
water  are  also  measured  by  the  height  of  a  column  of  mercury  or 
water.  Where  great  accuracy  is  required,  as  in  the  barometer,  a 
vernier  is  commonly  used. 

The  instrument  known  as  the  cathetometer  is  so  much  used  for 
measuring  heights  that  it  needs  a  notice  here.  It  consists  of  a 
small  telescope,  capable  of  sliding  up  and  down  a  vertical  rod 
to  which  a  scale  is  attached.  The  difference  in  height  of  any  two 
objects  is  readily  obtained  by  bringing  the  telescope  first  on  a  level 
with  one,  and  then  with  the  other,  and  taking  the  difference  in  the 
readings.  A  level  should  be  attached  to  the  telescope  to  keep  it 
always  horizontal,  but  the  great  objection  to  the  instrument  is  that 
a  very  slight  deviation  in  its  position,  which  may  be  caused  by 
focussing  or  turning  it,  is  greatly  magnified *in  a  distant  object.  A 
good  substitute  for  this  instrument  may  be  made  by  attaching  a 
common  telescope  to  a  vertical  brass  tube,  the  scale  being  placed 
near  the  object  to  be  measured  instead  of  on  the  tube,  as  in  Ex- 
periment 12. 

Although  the  measurement  of  the  following  quantities  is  directly 
dependent  on  the  above,  yet  their  importance  justifies  a  separate 
notice. 

Measurement  of  Areas.  It  is  difficult  in  general  to  determine 
an  area  with  accuracy,  especially  where  it  forms  the  boundary  of  a 
curved  surface.  If  plane,  any  of  the  methods  of  mensuration  used 
in  surveying  may  be  adopted.  Of  these  the  best  are  division  into 
triangles,  Simpson's  rule,  and  drawing  the  figure  on  rectangular 
paper  and  counting  the  number  of  enclosed  squares,  allowing  for 
the  fractions.  Another  method  sometimes  useful  is  to  cut  the  figure 
out  of  sheet  lead,  tin  foil,  or  even  card  board,  and  compare  its 
weight  with  that  of  a  square  decimetre  of  the  same  material. 

Measurement  of  Volumes.  These  are  generally  determined  by 
the  weight  of  an  equal  bulk  of  water  or  mercury,  using  the  latter 
if  the  space  is  small.  The  interior  capacity  of  a  vessel  is  meas- 
ured by  weighing  it  first  when  empty,  and  then  when  filled  with 
the  liquid,  as  in  Experiment  19.  The  difference  in  grammes 
gives  the  volume  in  cubic  centimetres  when  water  is  used,  but 
with  mercury  we  must  divide  by  13.6,  its  specific  gravity.  In  the 
same  way  we  may  determine  the  exterior  volume  of  any  body  by 


ANGLES.  23 

immersing  it  and  measuring  its  loss  of  weight,  as  when  determin- 
ing its  specific  gravity. 

An  easier,  but  less  accurate,  method  is  by  a  graduated  vessel. 
These  are  made  by  adding  equal  weights  or  volumes  of  liquid, 
successively,  and  marking  the  height  to  which  it  rises  after  each 
addition.  The  volume  of  any  space  may  then  be  found  by  filling 
it  with  water,  emptying  it  into  the  graduated  vessel  and  reading 
the  scale  attached  to  the  side  of  the  latter. 

Measurement  of  Angles.  Angles  are  measured  by  a  circle  di- 
vided into  equal  parts,  the  small  divisions  being  determined  by 
verniers  or  reading  microscopes,  as  in  measuring  lengths.'  A 
great  difficulty  arises  from  the  centre  of  the  graduation  not  coin- 
ciding with  that  of  the  circle,  and  on  this  account  it  is  best  to 
have  two  or  more  at  equal  intervals  around  the  circumference.  By 
taking  their  mean  we  eliminate  the  eccentricity. 

The  precision  of  modern  astronomy  is  almost  entirely  due  to 
the  methods  of  determining  angles  with  accuracy.  This  is  de- 
pendent on  two  things;  first,  a  good  graduated  circle,  and  sec- 
ondly, a  means  of  pointing  a  telescope  in  a  given  direction,  as 
towards  a  star,  with  great  exactness.  The  latter  is  accomplished 
by  placing  cross-hairs  at  the  common  focus  of  the  object  glass 
and  eye-piece,  so  that  they  may  be  distinctly  seen  in  the  centre  of 
the  field  at  the  same  time -as  the  object.  Most  commonly  two 
cross-hairs  are  used  at  right  angles,  one  being  horizontal,  the  other 
vertical.  When,  however,  we  are  to  bring  them  to  coincide  with 
a  straight  line,  as  in  the  spectroscope,  or  in  a  reading  microscope, 
they  are  sometimes  inclined  at  an  angle  of  about  60°,  that  is,  each 
making  an  angle  of  30°  with  the  line  to  be  observed.  The  latter 
is  then  brought  to  the  point  of  the  V  formed  by  their  intersection. 
Still  another  method  is  to  use  two  parallel  lines  very  near  together, 
the  line  to  be  observed  being  brought  midway  between  them. 
The  lines  may  be  made  of  the  thread  of  a  spider,  of  filaments  of 
silk,  of  platinum  wire,  or  better  for  most  purposes,  by  ruling  fine 
lines  on  a  plate  of  thin  glass  with  a  diamond,  and  inserting  it  at 
the  focus. 

There  are  two  methods  of  graduating  circles  with  accuracy. 
The  first,  which  is  used  in  Germany,  consists  in  a  direct  compari- 
son with  an  accurately  divided  circle,  as  when  copying  scales  as 


24  GRADUATING    CIRCLES. 

described  above.  That  is,  both  circles  are  mounted  on  the  same 
axis,  and  the  divisions  of  the  first  being  successively  brought  un- 
der the  cross-hairs  of  a  microscope,  the  graver  cuts  lines  on  the 
second  at  precisely  the  same  angular  intervals.  In  the  second 
method,  which  is  much  quicker  but  less  accurate,  the  circle  is  laid 
on  a  toothed  wheel  which  is  turned  through  equal  intervals  by 
a  tangent  screw.  Both  methods  are  really  only  means  of  copy- 
ing an  originally  divided  circle,  as  it  is  called,  and  the  con- 
struction of  this  with  accuracy  is  a  matter  of  extreme  difficulty. 
It  is  dependent  on  the  following  principles.  Any  arc  or  distance 
may  be  accurately  bisected  by  beam  compasses ;  the  chord  of  60° 
equals  the  radius,  and  the  angle  85°  20',  whose  chord  is  1.3554,  by 
ten  bisections  is  reduced  to  5'.  By  constructing  an  accurate  scale, 
laying  off  1.3554  times  the  radius  on  the  circumference,  and 
repeatedly  bisecting  the  arc,  we  finally  divide  the  circle  into 
5'  divisions.  Where  great  accuracy  is  not  required  we  may 
divide  circles  approximately  by  hand,  as  described  under  the 
Graphical  Method,  or  more  accurately  by  a  table  of  chords  and  a 
pair  of  beam  compasses.  When  the  divisions  of  the  circle  are 

very  large  we  may  subdivide  them  by 
123  a  scale  instead  of  a  vernier.     Thus  if 

AJ 1 — |NI|M[    I LB     ^g,Fig.4,  is  part  of  a  circle  divided 

J?  30  J,  into  degrees,  we  may  attach  a  scale 

Fig  4m  CD,  divided  to  ten  minutes,  and  sub- 

divide  these   into   single   minutes  by 

the  eye.     Thus  in  Fig.  4  the  reading  is  2°  35'.    Much  labor  is  thus 
saved  where  the  circles  have  to  be  divided  by  hand. 

Saxton's  pyrometer,  described  above,  is  of  the  utmost  value  in 
measuring  small  angular  changes.  As  the  reflected  beam  moves 
twice  as  fast  as  the  mirror,  the  accuracy  is  doubled  on  this  account. 
If  the  scale  is  flat,  allowance  must  be  made  for  the  greater  distance 
of  its  ends  than  the  centre.  To  reduce  the  reading  to  degrees  and 
minutes,  the  formula,  tan  2a  =  s  -f-  d  is  used,  or  a  =  .5  tan~ * 
s  -r-  df,  in  which  a  is  the  angle  through  which  the  mirror  turns,  s 
the  reading,  and  d  the  distance  of  the  scale  taken  in  the  same 
units.  Instead  of  a  telescope  a  light  shining  through  a  narrow  slit 
is  sometimes  used,  and  an  image  projected  on  the  scale  by  a  lens, 
or  the  mirror  itself  may  be  made  concave.  This  plan  is  adopted 


RADIUS    OF    CURVATURE.  25 

in  the  Thomson's  Galvanometer,  and  other  instruments  for  meas- 
uring the  deviations  of  the  magnetic  needle. 

Very  small  angles  may  also  be  measured  by  a  spider  line  mi- 
crometer attached  to  the  eye-piece  of  a  telescope.  This  is  used  to 
determine  the  distance  apart  of  the  double  stars,  and  other  minute 
astronomical  magnitudes.  There  are  other  methods,  such  as  di- 
vided lenses,  double  image  prisms,  &c.,  but  they  will  be  considered 
in  connection  with  the  particular  experiments  which  serve  to  illus- 
trate them. 

Measurement  of  Curvature.  To  measure  the  radius  of  a  sphere, 
as  the  surface  of  a  lens,  an  instrument  called  the  spherometer  is 
used.  It  consists  of  a  micrometer  screw  at  the  centre  of  a  tripod, 
whose  three  legs  and  central  point  are  brought  in  contact  with  the 
surface.  By  noting  the  position  of  the  screw,  the  radius  is  readily 
computed,  as  in  Experiment  14. 

When  the  surface  is  of  glass,  and  the  curvature  very  slight,  a 
much  more  delicate  method  is  as  follows :  Focus  a  telescope  on  a 
distant  object,  and  then  view  the  image  reflected  in  the  surface  to 
be  tested.  If  the  latter  is  concave,  it  will  render  the  ray  less  di- 
vergent, and  hence  the  eye-piece  will  have  to  be  pushed  in.  The 
opposite  effect  is  produced  by  a  convex  mirror.  The  amount  of 
change  affords  a  rough  measure  of  the  curvature.  This  method  is 
so  delicate  as  to  show  a  curvature  whose  radius  is  several  miles. 


GENERAL  EXPERIMENTS. 


1.    ESTIMATION  OF  TENTHS. 

Apparatus.  Two  scales,  N  and  j&f,  are  placed  side  by  side,  one 
being  divided  into  millimetres,  the  other  into  tenths  of  an  inch. 
Also  a  steel  rule  A,  Fig.  5,  divided  into  millimetres,  and  so  ar- 
ranged that  it  may  be  pushed  past  a  fixed  index  .Z?,  by  a  microm- 
eter screw,  C.  A  spring,  D,  is  used  to  bring  it  back,  when  the 
screw  is  turned  the  other  way. 

Experiment.  Read  the  position  of  each  tenth  of  an  inch  mark 
of  scale  M,  in  tenths  of  a  millimetre,  estimating  the  fractions  by 
the  eye.  Thus  if  the  interval  is  one  half,  call  it  .5,  if  a  little  less, 
.4,  if  not  quite  a  third,  .3,  and  so  on  for  the  other  fractions.  The 
.3  and  .7  are  the  hardest  to  estimate  correctly,  as  we  are  liable  to 
imagine  the  former  too  great,  the  latter  too  small.  They  should 
always  be  compared  with  the  fractions  one  and  two  thirds.  Re- 
cord your  observations  in  five  columns,  placing  in  the  first  the 
readings  of  the  scale  M, 
in  the  second  the  corre- 
sponding readings  of  N, 
and  in  the  third  the  first 
differences  of  N.  Next, 
subtract  the  first  from  the 
last  number  in  column  two, 


J 


and  divide  the  difference 

by  the   number  of  spaces 

measured,  that  is,  the  num- 

ber of  readings  minus  one. 

This  gives  the  average  dif- 

ference, and  should  be  equal  to  each  number  of  golurnn   three. 

Subtract  it  from  these  numbers,  and  place  the  results  or  errors, 

with  proper  signs,  in  column  four.    Next,  compute  the  probable 


Fig.  5. 


28  ESTIMATION    OF    TENTHS. 

error  (see  page  3)  of  a  single  observation,  using  the  fifth  column 
for  the  squares  of  column  four.  In  this  way  you  can  read  any 
scale  much  more  accurately  than  by  its  single  divisions,  and  your 
computed  probable  error  shows  how  closely  you  may  rely  on  the 
result. 

Next  bring  one  of  the  millimetre  marks  of  A,  Fig.  5,  opposite 
the  index  B.  Read  its  position,  as  described  on  page  20.  The 
scale  E  gives  units,  or  number  of  revolutions,  and  the  divided 
circle  hundredths.  Move  the  screw,  set  again,  and  repeat  several 
times.  Take  the  mean  and  compute  the  probable  error  of  a  single 
observation.  Do  the  same  with  the  next  millimetre  mark.  Now 
move  the  scale  until  the  reading  shall  be  in  turn  .1,  .2,  .3,  &c., 
of  a  millimetre,  taking  care  to  move  the  screw  after  each,  so  that 
you  will  not  be  biassed  by  your  previous  reading.  Next  compute 
what  should  be  the  true  readings  in  these  various  positions.  Thus 
let  m'  be  the  mean  for  the  first  millimetre,  m"  for  the  second  ;  the 
reading  for  one  tenth  would  be  mf  +  (m"  —  m')  -f-  10,  for  two 
tenths  m'  +  2(m"  —  m')  -r- 10,  and  so  on.  See  how  these  read- 
ings agree  with  those  previously  found.  If  any  differ  by  a  consid- 
erable amount  repeat  them  until  you  can  estimate  any  fraction 
with  accuracy.  This  work  must  be  carefully  distinguished  from 
guessing,  since  there  should  be  no  element  of  chance  in  it,  but  an 
accurate  division  of  the  spaces  by  the  eye.  By  practice  one  can 
read  these  fractions  almost  as  accurately  as  by  a  vernier. 

2.    VERNIERS. 

Apparatus.  A  number  of  verniers  and  scales  along  which  they 
slide  are  made  of  large  size.  The  best  material  is  metal  or  wood,- 
although  cardboard  will  do.  By  making  them  on  a  large  scale,  as 
a  foot  or  more  in  length,  there  is  no  trouble  in  attaining  sufficient 
accuracy.  Several  different  forms  are  given  in  Gillespie's  Land 
Surveying,  p.  228,  from  which  the  following  may  be  selected. 

1st,  Fig.  225,  Scale  divided  to  .1,  Vernier  reads  to  .01 ;  2d,  Fig. 
227,  Same  Vernier  retrograde ;  3d,  Fig.  228,  Scale  .05 ;  Vernier 
.002;  4th,  Fig.  229,  Scale  1°,  Vernier  5';  5th,  Fig.  230,  Scale  30', 
Vernier  1';  .6th,  Fig.  233,  scale  20',  Vernier  30";  7th,  Fig.  239,' 
Scale  30',  Vernier  1' ;  Double  Compass  Vernier. 

Experiment.  A  vernier  may  be  regarded  as  a  simple  enlarge- 
ment of  one  division  of  the  scale.  Thus  if  the  scale  is  divided 


INSERTION    OF    CROSS-HAIRS.  29 

into  tenths  of  an  inch,  and  the  vernier  into  ten  parts,  it  will  read 
to  hundredths  of  an  inch.  Always  read  approximately  by  the 
zero  of  the  vernier,  taking  the  division  of  the  scale  next  below  it. 
The  fraction  to  be  added  is  found  by  seeing  what  line  of  the  vernier 
coincides  most  nearly  with  some  line  of  the  scale.  Thus  in  the 
first  example,  we  obtain  inches  and  tenths  by  seeing  what  division 
of  the  scale  falls  next  below  the  zero  of  the  vernier.  If  this  is  8.6, 
and  the  division  marked  7  of  the  vernier  coincides  with  a  line  of 
the  scale,  the  true  reading  is  8.6  +  .07  =  8.67.  To  prove  this,  set 
the  zero  of  the  vernier  at  8.6  exactly.  Nine  divisions  of  the  scale 
equal  ten  of  the  vernier.  Hence  each  division  of  the  latter  equals 
.09,  or  is  shorter  by  .01  than  one  division  of  the  scale.  Accord- 
ingly the  line  marked  1  of  the  vernier  falls  short  by  .01  of  the 
scale-division,  the  2  line  .02,  and  so  on.  If  we  move  the  vernier 
forward  by  these  amounts  these  lines  will  coincide  in  turn.  Hence 
when  the  7  line  coincides,  as  in  the  above  example,  it  denotes  that 
the  vernier  has  been  pushed  forward  .07  beyond  the  8.6  mark. 
This  method  may  be  applied  to  reading  any  vernier.  To  find  the 
magnitude  of  the  divisions  of  the  latter,  divide  one  division  of  the 
scale  by  the  number  of  parts  contained  in  the  vernier. 

Read  and  record  the  verniers  as  now  set.  Then  set  them  as 
follows:  1st,  8.03;  2d,  29.9;  3d,  30.866;  4th,  4°  10' 5  5th,  0°  17'; 
6th,  2°58/30";  7th,  2°  51'. 

The  last  vernier  is  a  double  one,  reading  either  way,  the  left 
hand  upper  figures  being  the  continuation  of  those  on  the  lower 
right  hand.  This  is  best  understood  by  moving  it  5'  at  a  time  and 
noting  what  lines  coincide. 

After  each  exercise  the  instructor  should  set  all  the  verniers,  and 
compare  the  record  of  the  student  with  his  own. 

3.    INSERTION  OP  CROSS-HAIRS. 

Apparatus.  Some  common  sewing  silk,  card-board  and  muci- 
lage, also  a  pair  of  dividers,  ruler  and  triangle. 

Experiment.  A  great  portion  of  the  accuracy  attained  in  mod- 
ern astronomical  work  is  dependent  on  the  exactness  with  which 
we  can  point  a  telescope,  or  other  similar  instrument,  in  a  given 
direction.  This  is  accomplished  by  inserting  two  filaments  of  silk 


30 


INSERTION    OF    CROSS-HAIRS. 


or  spider's  web  at  right  angles  to  each  other,  at  the  point  within 
the  telescope  where  the  image  of  the  object  is  formed.  In  the 
astronomical  telescope,  where  a  positive  eye-piece  is  used,  this 
point  lies  just  beyond  the  eye-piece,  that  is  between  it  and  the 
object-glass.  A  ring  is  placed  at  this  point  on  which  the  lines  are 
stretched.  In  telescopes  rendering  objects  upright,  as  in  most 
surveyor's  transits,  the  lines  are  commonly  placed  between  the 
object-glass  and  erecting  lenses,  and  close  to  the  latter.  In  the 
microscope,  and  other  instruments  where  a  negative  eye-piece  only 
is  used,  the  lines  have  to  be  placed  on  the  diaphragm  between  the 
field-  and  eye-lenses.  This  plan  is  objectionable,  since  the  lines 
should  be  very  accurately  focussed,  which  can  then  only  be  done 
by  screwing  the  eye-lens  in  or  out.  In  the  other  cases  the  whole 
eye-piece  maybe  slid  in  or  out  until  the  lines  are  perfectly  distinct, 
and  do  not  appear  to  move  over  the  object  when  the  eye  is  moved 
from  side  to  side. 

It  is  comparatively  easy  to  insert  the  lines  on  their  ring,  where 
a  positive  eye-piece  is  used.  The  following  experiment  therefore 
includes  the  others.  Take  a  negative  eye-piece, 
Fig.  6,  from  a  microscope  or  telescope,  and  unscrew 
the  eye-lens  A.  C  is  the  diaphragm  which  limits 
the  field  of  view,  and  on  which  the  lines  should  be 
placed.  Cut  from  the  cardboard  a  ring,  Fig.  7, 
whose  inner  diameter  is  a  little  greater  than  the 
opening  of  the  diaphragm,  and  the  outer  diameter 
such  that  it  will  easily  rest  on  C.  Mark  on  it  two 


Fig.  6. 


lines  at  right  angles  to  each  other  passing  through  its  centre.  Un- 
ravel a  short  piece  of  the  silk  thread  until  you  have  separated  a 
single  filament.  This  is  best  done  by  holding 
the  thread  with  the  forceps  over  a  sheet  of  white 
paper.  We  now  wish  to  stretch  two  of  these 
filaments  over  the  lines  marked  on  the  card- 
board circle.  Put  a  little  mucilage  on  the  lat- 
ter, dip  one  end  of  the  silk  into  it,  and  press  it 
down  with  one  of  the  radial  strips  of  paper 
shown  in  Fig.  7.  When  this  is  nearly  dry  fasten  the  other  end  in 
the  same  way,  taking  care  to  stretch  it  so  that  it  shall  be  straight, 
or  the  twist  in  the  thread  will  give  it  a  sinuous  form.  Attach  the 


Fig.  7. 


SUSPENSION    BY    SILK    FIBRES.  31 

other  thread  in  the  same  way,  and  bending  the  four  strips  of  paper 
down  lay  the  cardboard  on  the  diaphragm.  To  hold  it  in  place 
cut  a  strip  of  cardboard  or  brass,  and  bending  it  into  a  circle  push 
it  into  the  tube.  By  its  elasticity  it  will  hold  the  paper  strips 
firmly  against  the  sides  of  the  tube.  If  the  experiment  has  been 
well  performed,  on  replacing  the  eye-lens  we  see  two  straight  lines 
at  right  angles,  dividing  the  field  of  view  into  four  equal  parts. 
The  cardboard  should  not  project  beyond  the  diaphragm,  or  it  will 
give  a  rough  edge  to  the  field  of  view,  and  we  must  be  careful  that 
no  mucilage  adheres  to  the  visible  portions  of  the  threads. 

4.     SUSPENSION  BY  SILK  FIBRES. 

Apparatus.  The  best  method  of  suspending  a  light  object  so 
that  it  shall  move  very  freely  is  by  a  single  filament  of  silk.  The 
only  apparatus  needed  is  a  stand  seven  or  eight  inches  high,  some 
unspun  silk  (common  silk  thread  will  do,  but  is  not  so  good)  and 
some  fine  copper  wire.  We  also  need  two  pairs  of  forceps,  such  as 
come  with  cheap  microscopes,  some  bees-wax  and  a  sheet  of  white 
paper. 

Experiment.    Lay  the  silk  on  the  paper  and  pick  out  a  single 
fibre  a  little  over  six  inches  long.     Bend  pieces  of  the  wire  into 
the  shapes  A  and  J?,  Fig.  8.     Pass  one  end  of  the 
filament  through  the  ring  of  J?,  and  fasten  it  with    .  P  % 

a  little  wax,  twisting  or  tying  it  to  prevent  slip-  "          u 

ping.    Fasten  the  other  end  to  A  in  the  same  way,  Fi    g 

making  the  distance  from  A  to  B  just  six  inches. 
Hook  A  into  the  stand,  and  lay  the  object  to  be  suspended,  as  a 
needle  on  IB. 

5.    TEMPERATURE  CURVE. 

Apparatus.  A  beaker,  stand  and  burner,  by  which  water  can 
be  heated,  a  Centigrade  thermometer,  and  a  clock  or  watch  giving 
seconds. 

Experiment.  Place  the  thermometer  in  the  water  and  record 
the  temperature,  dividing  the  degrees  to  tenths,  as  described  in 
Experiment  1.  Place  the  burner  under  the  beaker  at  the  begin- 
ning of  a  minute,  and  at  the  end  record  the  temperature ;  repeat 
at  the  end  of  each  minute,  as  the  water  is  warmed,  until  the  ther- 


32 


TESTING     THERMOMETERS. 


mometer  stands  at  95° ;  at  the  end  of  the  next  minute  remove 
the  burner  and  the  temperature  will  at  first  continue  to  rise,  and 
will  then  fall  rapidly.  Record  the  time  (in  minutes  and  seconds) 
of  attaining  95°,  90°,  85°,  &c.,  taking  shorter  intervals  as  the  tem- 
perature becomes  lower,  and  the  cooling  less  rapid.  Record  your 
results  in  two  columns,  one  giving  times,  the  second  temperatures. 
Finally  construct  a  curve  in  which  abscissas  represent  times,  and 
ordinates  temperatures,  making  in  the  former  case,  one  space  equal 
one  minute,  in  the  latter,  one  degree. 

When  two  students,  A  and  _Z?,  are  engaged  in  this  experiment, 
the  following  system  should  be  used.  A  observes  the  watch  and 
records,  while  B  attends  to  the  thermometer.  Five  seconds  be- 
fore the  minute  begins  A  says,  Heady !  and  at  the  exact  begin- 
ning, Now!  JB  then  gives  the  reading  which  A  record's.  This 
plan  saves  much  trouble,  and  greatly  increases  the  accuracy  of  any 
observations  which  must  be  made  at  regular  intervals  of  time. 

6.     TESTING  THERMOMETERS. 

Apparatus.  An  accurate  Centigrade  thermometer  is  hung  upon 
a  stand,  and  close  to  it  a  Fahrenheit  thermometer,  which  is  to  be 
tested,  their  bulbs  being  at  the  same  height,  and  close  together. 
A  telescope  with  which  they  can  be  read  more  accurately  is  placed 
on  a  stand  at  a  short  distance,  and  their  temperature  may  be  al- 
tered at  will  by  immersing  their  bulbs  in  a  beaker  of  water,  which 
may  be  either  cooled  by  ice,  or  heated  by  a  Bunsen  burner.  Some 
arrangement  is  desirable  for  stirring  the  water  to  keep  it  at  a  uni- 
form temperature.  One  way  is  to  use  a  circular  disk  of  tin  with 
holes  cut  in  it,  which  may  be  raised  or  lowered  in  the  beaker  by  a 
cord  passing  over  a  pulley,  so  that  the  observer,  while  looking 
through  the  telescope,  can  stir  the  water  by  alternately  tightening 
and  loosening  the  cord.  A  simple  glass  stirring  rod  may  be  used 
instead,  if  preferred. 

Experiment.  The  problem  is  to  determine  the  error  of  the 
Fahrenheit  thermometer  at  different  temperatures,  by  comparing 
it  with  the  Centigrade  thermometer,  which  is  regarded  as  a  stand- 
ard. By  means  of  the  telescope  read  them  as  they  hang  in  the 
air,  estimating  the  fractions  of  a  degree  in  tenths.  Do  the  same 
when  their  bulbs  are  immersed  in  water,  then  cool  them  with  ice 
and  read  again.  This  observation  is  important,  as  it  shows  the 
absolute  error  of  each  instrument.  Next  heat  the  water  a  few 


ECCENTRICITY    OF    GRADUATED    CIRCLES.  33 

degrees  with  the  burner,  and  then  remove  the  latter.  The  tem- 
perature will  still  rise  for  a  short  time,  then  become  stationary  and 
fall.  Read  each  thermometer  at  its  highest  point,  stirring  the  wa- 
ter meanwhile.  Repeat  at  intervals  of  about  10°  until  the  water 
boils,  and  finally  immerse  again  in  the  ice  water,  and  see  if  the 
reading  is  the  same  as  before. 

We  have  now  two  columns  of  figures,  the  first  giving  the  tem- 
perature of  the  Centigrade,  the  second  that  of  the  Fahrenheit, 
thermometer.  Reduce  the  first  to  the  second,  recollecting  that 
0°  C.  =  32°  F.,  and  100°  C.  =  212°  P.;  hence  F.  =  f  C.  +  32°, 
calling  C  and  F  the  corresponding  temperatures  on  the  Centi- 
grade and  Fahrenheit  scales  respectively.  Write  the  numbers 
thus  found  in  a  third  column,  and  the  errors  will  equal  the  differ- 
ences between  them  and  the  readings  given  in  column  two.  If 
the  Centigrade  thermometer  does  not  stand  at  zero  when  im- 
mersed in  ice  water,  all  its  readings  should  be  corrected  by  the 
amount  of  the  deviation,  taking  care  to  retain  the  proper  sign. 
Now  construct  a  curve  whose  ordinates  shall  represent  the  errors 
on  an  enlarged  scale,  and  abscissas  the  temperatures. 

7.    ECCENTRICITY  OF  GRADUATED  CIRCLES. 

Apparatus.  A  circle  divided  into  degrees  carries  a  pointer 
with  an  index  at  each  end,  which  turns  eccentrically,  that  is,  the 
centres  of  the  pointer  and  circle  do  not  coincide.  It  may  be  made 
in  a  variety  of  ways.  One  of  the  simplest  is  to  place  a  pivot  on 
one  side  of  the  centre  of  the  circle,  and  on  it  a  rod  with  a  needle 
projecting  from  each  end.  Another  way  is  to  let  the  circle  turn 
and  cover  it  with  a  plate  of  glass,  on  which  are  marked  two  fine 
lines,  with  a  diamond  or  India  ink.  The  indices  may  also  be  made 
of  fine  wire,  or  horsehair.  Lines  of  consid- 
erable length  must  be  used,  since  the  edge 
of  the  circle  advances  and  recedes  as  it  is 
turned.  If  greater  accuracy  is  desired  the 
plan  shown  in  Fig.  9  may  be  adopted.  The 
two. indices  (which,  may  have  verniers)  are 
connected  with  the  centre  by  the  arms  A  G 
and  CB.  The  circle  turns  around  the  pin 
J9,  and  a  rod  passing  through  the  guides 
EF,  keeps  the  verniers  in  the  proper  posi- 
tion. Another  good  instrument  for  this  experiment  is  the  form  of 
compass  described  under  Magnetism  in  the  latter  part  of  the  pres- 
ent work. 


34  CONTOUR   LINES. 

Experiment.  Set  the  index  A  at  0°  by  turning  the  circle,  and 
read  £.  Repeat  moving  A  10°  at  a  time,  until  a  complete  revo- 
lution has  been  made.  We  have  now  two  columns,  giving  the 
corresponding  readings  of  A  and  JB.  Subtract  180°  from  the  lat- 
ter, and  \(A  +  B  —  180°),  or  \(A  +  JB)  —  90°  will  be  the  true 
reading ;  write  this  in  column  three ;  in  the  same  way  the  error  of 
each  index  is  \(A  — B)  —  90°,  which  should  be  written  in  the 
fourth  column.  Construct  a  curve  with  abscissas  equal  to  the 
numbers  in  column  three,  and  ordinates  equal  to  those  in  column 
four,  enlarged.  At  the  highest  and  lowest  parts  of  the  curve  the 
indices  differ  most  from  their  true  position,  or  the  absolute  error, 
if  we  read  one  only,  is  here  greatest.  Find  these  points  by  Curves 
of  Error,  p.  14.  On  the  other  hand,  where  the  curve  cuts  the  axis 
the  two  indices  are  opposite  each  other,  and  the  abscissa  gives  the 
azimuth  of  the  line  CD.  As  the  ordinates  alter  most  rapidly  at 
these  points,  the  error,  when  reading  a  small  angle  by  one  index,  is 
here  a  maximum.  Draw  tangents,  as  before,  by  Curves  of  Error, 
and  from  their  direction  we  can  compute  the  amount  of  variation. 
It  is  a  very  good  exercise  to  deduce  by  trigonometry  the  theoreti- 
cal curve,  and  constructing  it  on  the  same  sheet  of  paper  to  com- 
pare the  results  with  those  obtained  by  your  measurement. 

We  have  heretofore  supposed  that  the  line  connecting  the  in- 
dices passed  through  the  axis  around  which  they  turned,  or  that 
D  lies  on  EF.  If,  as  often  happens  in  practice,  this  is  not  the 
case,  a  second  correction  is  necessary. 

8.     CONTOFK  LINES. 

Apparatus.  No  apparatus  is  needed  for  this  experiment,  except 
ordinary  writing  materials.  It  is,  in  fact,  an  exercise  rather  than 
an  experiment. 

Experiment.  Mark  in  your  note  book  nine  rows  of  six  points 
each,  so  as  to  form  forty  squares  of  about  one  inch  on  a  side. 
Mark  them  with  numbers  taken  from  the  adjoining  table  A.  Now 
suppose  these  numbers  represent  the  heights  of  the  points  to  which 
they  are  attached,  and  we  wish  to  draw  contour  lines  to  show  the 
form  of  the  surface  passing  through  them.  As  the  points  are 
pretty  near  together  we  may  assume  that  a  line  connecting  any 


CLEANING    MERCURY. 


35 


two  that  are  adjacent  will  lie  nearly  in  the  surface.    Now  regard 
your   drawing   as   a   map,  as   on  p.  15,  and  suppose  the   ground 


B 


83 

79 

73 

79 

79 

74 

46 

56 

67 

84 

86 

84 

66 

76 

68 

57 

40 

28 

82 

78 

70 

81 

84 

76 

29 

52 

73 

94 

'86 

73 

73 

72 

50 

29 

28 

52 

78 

76 

66 

83 

88 

73 

39 

31 

65 

82 

.70 

56 

66 

48 

31 

11 

27 

41 

74 

73 

58 

78 

82 

63 

60 

62 

68 

72 

57 

49 

59 

29 

29 

42 

38 

29 

70 

61 

50 

73 

82 

74 

69 

73 

81 

81 

65 

48 

38 

46 

38 

72 

61 

39 

71 

58 

61 

82 

96 

75 

80 

94 

80 

81 

73 

50 

27 

35 

70 

99 

70 

28 

70 

59 

70 

83 

84 

72 

80 

58 

58 

65 

70 

49 

21 

46 

87 

96 

60 

29 

67 

65 

72 

79 

73 

69 

67 

58 

58 

67 

62 

46 

33 

63 

95 

81 

49 

31 

66 

67 

72 

76 

69 

75 

74 

68 

72 

80 

49 

37 

44 

71 

86 

64 

47 

27 

flooded  with  water  to  a  height  of  80.  Evidently  all  the  points  in 
the  upper  line  will  be  submerged  except  that  on  the  left,  and  the 
shore  line  will  come  between  79  and  83,  about  a  fourth  way  from 
the  former.  Also  midway  between  82  and  78  in  the  second  line, 
two  fifths  of  the  way  from  78  to  83,  and  a  third  way  from  79  to 
82.  Several  points  are  thus  obtained  in  each  square  through  which 
the  contour  line  passes.  After  obtaining  as  many  as  possible, 
draw  a  smooth  curve  nearly  coinciding  with  them  all,  paying 
special  attention  to  the  rules  given  under  the  Graphical  Method. 
Construct  in  the  same  way  other  contours  at  intervals  of  ten  units. 
Do  the  same  with  the  numbers  in  table  B  or  C. 

This  work  is  very  well  supplemented  by  procuring  from  the 
U.  S.  Signal  Bureau  at  Washington,  some  of  their  blank  maps 
(issued  at  $2.75  per  100),  and  filling  them  out  from  the  weather 
reports  for  the  day,  according  to  their  published  directions.  These 
maps  may  also  be  used  for  drawing  isothermals,  isogonals,  &c.,  if  a 
list  is  prepared  in  the  first  place  of  the  temperature,  magnetic 
variation,  &c.,  of  a  large  number  of  stations  in  the  United  States. 
The  method  adopted  for  drawing  these  lines  is  essentially  the  same 
as  that  given  above,  only  the  points  are  irregularly  spaced. 

9.     CLEANING  MERCURY. 

Apparatus.  But  little  apparatus  is  needed  for  this  experiment, 
except  such  as  is  found  in  every  chemical  laboratory.  Some  bot- 
tles, funnels,  &c.,  should  be  placed  on  the  table,  and  the  student 
should  try  as  many  of  the  following  methods  of  purification  as  he 
can,  and  record  in  his  note-book  his  opinion  of  their  comparative 
value. 


36  CLEANING    MERCURY. 

Experiment.  Mercury  is  so  much  used  in  physical  experiments 
that  every  student  should  know  how  to  clean  it.  The  impurities 
may  be  divided  into  three  classes :  first,  mixture  with  metals,  es- 
pecially lead,  zinc  and  tin  ;  secondly,  common  dust  and  dirt;  and 
thirdly,  water  or  other  liquids. 

Redistillation  is  almost  the  only  way  to  remove  the  metals,  and 
even  this  is  not  perfectly  effectual,  especially  in  the  case  of  zinc. 
Moreover,  by  long  boiling  a  small  amount  of  oxide  is  formed, 
which  is  dissolved  by  the  metal.  The  mercury  used  for  amal- 
gamating battery  plates  should  therefore  be  kept  separate  from  the 
rest  and  used  for  this  purpose  only.  If  but  little  of  the  metal  is 
present  it  may  be  removed  by  agitating  with  dilute  nitric  acid. 
The  best  way  to  do  this  is  to  fill  a  long  vertical  tube  with  the  acid 
and  allow  the  mercury  to  flow  into  it  from  a  funnel,  in  which  is  a 
paper  filter  with  a  fine  hole  in  the  bottom.  The  mercury  falls 
through  the  long  column  of  liquid  in  minute  globules,  and  is  thus 
readily  and  thoroughly  cleaned.  It  may  be  drawn  out  below  by 
a  glass  stopcock,  or  by  a  bent  tube  in  which  a  short  column  of 
mercury  shall  balance  a  long  column  of  acid.  As  the  mercury 
collects  it  flows  out  of  the  end  of  the  tube  into  a  vessel  placed  to 
receive  it.  Instead  of  nitric  acid  a  solution  of  nitrate  of  mercury 
may  be  used,  if  preferred.  Another  method  is  to  fill  a  bottle 
about  a  quarter  full  of  mercury,  add  a  quantity  of  finely  powdered 
loaf  sugar,  and  shake  violently.  The  metallic  impurities  are  ox- 
idized at  the  expense  of  the  air,  which  must  be  renewed  by  a  pair 
of  bellows. 

A  great  variety  of  devices  are  used  to  remove  the  mechanical 
impurities  of  mercury.  For  example,  pouring  it  into  a  bag  of 
chamois  leather  and  squeezing  the  latter  until  the  mercury  comes 
through  in  fine  globules.  Or,  making  a  needle  hole  in  the  point 
of  a  paper  filter,  placing  it  in  a  funnel  and  letting  the  mercury  run 
through.  The  mercury  may  be  washed  directly  with  water,  by 
shaking  them  together  in  a  bottle,  or  better,  filling  a  jar  half  full 
of  mercury  and  letting  the  water  from  the  hydrant  bubble  up 
through  it.  This  is  an  excellent  way  to  remove  most  liquids. 

Next,  to  remove  the  water,  pour  the  mixture  into  a  small  bottle, 
when  the  mercury  will  settle  to  the  bottom,  and  the  water  over- 
flow from  the  top.  When  the  mercury  fills  the  bottle  transfer  it 


CALIBRATION    BY    MERCURY.  37 

to  another  vessel  and  repeat.  If  there  is  only  mercury  enough  to 
half  fill  the  bottle  the  second  time,  pour  back  some  of  the  mercury 
already  dried  to  displace  the  remaining  water.  Another  way  is  to 
close  the  end  of  a  funnel  with  the  finger  and  pour  in  the  mixture, 
drawing  off  the  mercury  below  and  leaving  the  water  above.  Care 
must  be  taken  that  the  mercury  does  not  spurt  out  on  one  side 
and  escape.  An  inverted  bottle,  or  better,  a  vessel  with  a  tube 
and  stopcock  below,  is  more  convenient  for  this  purpose. 

When  only  a  few  drops  of  water  are  present  they  may  be  re- 
moved by  blotting  paper,  or  a  camel's  hair  brush.  Also  by  apply- 
ing heat ;  but  in  this  case  a  stain  will  be  left  when  the  water  evap- 
orates, unless  it  has  been  previously  distilled. 

To  see  if  the  mercury  is  pure  pour  it  into  a  porcelain  evaporat- 
ing dish.  If  lead  is  present  it  will  tarnish  the  sides.  A  thin  film 
will  also,  after  a  short  time,  form  on  its  surface,  due  to  oxidation  ; 
zinc  and  tin  produce  a  similar  effect.  The  surface  when  at  rest 
should  be  very  bright  and  almost  invisible,  and  small  globules,  if 
detached,  should  be  perfectly  spherical,  and  not  adhere  to  the  glass 
but  roll  over  it  when  the  surface  is  inclined. 

10.    CALIBRATION  BY  MERCURY. 

Apparatus.  The  best  way  to  perform  this  experiment  is  that 
given  by  Bun  sen  in  his  Gasometry,  p.  27.  This  method  is  sub- 
stantially as  follows :  Select  a  glass  tube,  about  2  cm.  in  diameter, 
and  40  cm.  long,  closed  at  one  end.  Fasten  to  it  a  paper  mil- 
limetre scale.  This  is  placed  upright  in  a  stand,  at  a  short  dis- 
tance from  a  small  telescope,  by  which  the  scale  may  be  read  with 
accuracy.  On  another  stand  is  placed  a  vessel  containing  about 
two  kilogrammes  of  pure  mercury,  covered  with  a  layer  of  concen- 
trated sulphuric  acid,  with  a  stopcock  below,  by  which  it  may  be 
drawn  off.  A  small  glass  tube,  also  closed  at  one  end,  is  used  to 
receive  it,  which  should  contain,  when  filled,  about  10  cm.8  Its 
open  end  is  ground  flat,  and  it  may  be  closed  with  a  plate  of 
ground  glass,  which  is  fastened  to  the  thumb  by  a  piece  of  rubber. 

Experiment.  Both  mercury  and  tube  should  be  perfectly  clean, 
but  if  not,  a  few  drops  of  water  may  be  placed  in  the  longer  tube, 
provided  great  accuracy  is  not  required.  Fill  the  small  tube  with 
mercury,  holding  it  with  the  fingers  of  the  left  hand,  and  remove 
the  surplus  by  pressing  the  glass  plate,  which  should  be  attached 
to  the  left  thumb,  down  on  to  it.  Take  care  that  no  air  bubbles 


38  CALIBRATION    BY    MERCURY. 

are  imprisoned.  Empty  the  mercury  into  the  large  tube,  and  read 
its  height  on  the  scale  by  the  telescope,  measuring  from  the  top 
of  the  curved  surface  of  the  liquid.  A  clean  wooden  rod  may  be 
used  to  remove  any  bubbles  of  air  or  globules  of  mercury  which 
adhere  to  the  sides  of  the  tube.  Repeat  this  operation  until  the 
large  tube  is  full  of  mercury.  We  now  wish  to  know  the  volume 
of  the  small  tube,  as  this  is  the  unit  in  terms  of  which  the  larger 
one  has  been  calibrated.  The  most  accurate  way  to  do  this  is  to 
weigh  the  whole  amount  of  mercury  transferred,  and  divide  by 
the  number  of  times  the  smaller  tube  has  been  filled.  But  as  it  is 
generally  difficult  to  weigh  so  heavy  a  body  accurately,  the  con- 
tents of  the  smaller  tube  had  better  be  weighed  alone,  repeating 
two  or  three  times  to  see  how  much  the  quantity  used  will  vary  in 
consecutive  fillings.  The  volume  is  then  obtained  by  dividing  the 
weight  by  13.6,  the  specific  gravity  of  mercury.  Multiplying  the 
quotient  by  1,  2,  3,  4,  &c.,  we  obtain  the  volumes  corresponding  to 
our  observed  readings  of  the  mercury  column  in  the  long  tube. 

Represent  the  results  by  a  residual  curve,  as  follows :  Let  s  be 
the  scale  reading  when  the  small  tube  has  been  emptied  once  into 
the  long  tube,  and  /  when  the  latter  is  full,  or  has  received  n  times 
this  volume  of  mercury,  which  we  will  call  v.  Then  (n  —  l)v  of 
mercury  will  fill  the  space  sf  —  s,  and  the  average  volume  per  unit 
of  length  will  equal  (n  —  l)u  -r-  (/  —  s)  =  a.  If  the  tube  was 
perfectly  cylindrical  we  could  find  the  volume  V  for  any  scale 
reading  S  by  the  formula,  V  =  a  {S  —  s)  -f  v.  In  reality  the 
tube  is  probably  a  little  larger  in  some  places  than  in  others, 
it  is  therefore  better  to  retain  only  two  significant  figures  in  a, 
and  then  compute  by  the  formula  the  volumes  corresponding  to 
the  various  scale  readings  that  have  been  observed.  Subtract 
each  of  these  from  the  corresponding  volumes  1,  2,  3,  &c.,  times  v, 
and  construct  a  residual  curve  in  which  ordinates  equal  these  dif- 
ferences on  an  enlarged  scale,  and  abscissas  the  scale  readings. 
We  can  now  obtain  the  volume  with  the  greatest  accuracy  for  any 
scale  reading  by  adding  to  the  value  of  V  given  by  the  formula, 
the  ordinate  of  the  corresponding  point  of  the  curve.  A  table 
may  thus  be  constructed,  giving  the  volume  corresponding  to  each 
millimetre  mark  of  the  scale.  But  it  is  generally  sufficiently  accu- 
rate to  make  a  simple  interpolation  from  the  original  measurements, 


CALIBRATION    BY   WATER.  39 

using  only  the  first  differences,  as  when  employing  logarithmic 
tables. 

11.     CALIBRATION  BY  WATER. 

Apparatus.  A  Mohr's  burette  B,  Fig.  10,  on  a  stand,  and  the 
vessel  to  be  graduated  A,  which  should  be  about  six  inches  high, 
and  an  inch  and  a  half  in  diameter.  A  paper  scale  divided  into 
tenths  of  an  inch  should  be  attached  to  A  with  gum  tragacanth, 
although  shellac,  or  even  mucilage,  answers  tolerably.  A  long 
string  wound  spirally  around  the  vessel  will  keep  the  scale  in  place 
until  the  gum  is  dry. 

Experiment.  Fill  the  burette  JB  to  the  zero  mark.  This  is 
done  by  adding  a  little  too  much  water,  and 
drawing  it  off  by  the  stopcock  C  into  another 
vessel,  until  it  stands  at  precisely  the  right  level. 
Next,  let  the  water  flow  into  A  until  it  reaches 
the  one  tenth  of  an  inch  mark,  and  read  J5.  Do 
the  same  for  each  tenth  of  an  inch,  until  the  one 
inch  mark  is  reached,  and  then  for  every  half 
inch  to  the  top.  Do  not  let  the  water  level  in  J3  :DQ 

^7 

fall    below   the    100    cm.3   mark,   but   when    it     ' — 
reaches  this  point  refill  as  before,  and  add  100  to  Fi£- 10- 

the  volume  measured.  Care  should  be  taken  not  to  get  too  much 
water  into  A ;  should  this  happen,  a  little  may  be  drawn  out  with 
a  pipette  and  replaced  in  J?,  but  a  slight  error  is  thus  introduced. 

We  have  now  a  series  of  volumes  corresponding  to  various  scale 
readings.  Construct  a  curve  with  these  two  quantities  as  co- 
ordinates. Find  the  point  of  the  curve  for  which  the  volume  is  in 
turn  10,  20,  30,  &c.,  cm.8,  and  record  the  corresponding  scale- 
reading.  If  the  vessel  is  to  be  used  for  the  measurement  of  vol- 
umes cover  it  with  wax  and  draw  horizontal  lines  on  the  latter, 
having  the  scale  readings  just  found.  Subject  it  to  the  fumes  of 
fluorhydric  acid,  formed  by  mixing  powdered  fluor  spar  and  con- 
centrated sulphuric  acid.  The  lines  will  thus  be  permanently 
etched  on  the  glass. 

12.  CATHETOMETER. 

Apparatus.  A  Cathetometer  may  be  made  by  using  as  a  base 
the  tripod  of  a  music  stand  or  photographer's  head-rest,  and  screw- 


40 


CATHETOMETER. 


ing  into  it  a  tube  or  solid  rod  of  brass.  To  this  is  attached  a  small 
telescope  with  a  clamp  and  set  screw,  and  some  form  of  slow 
motion.  The  latter  may  be  obtained  by  placing  the  telescope  on 
a  hinge  and  raising  and  lowering  one  end  by  a  screw.  The  slight 
deviation  from  a  horizontal  position  will  not  affect  the  results,  as 
the  instrument  is  here  used. 

At  a  distance  of  five  or  ten  feet  is  placed  a  U  tube,  open  at  both 
ends,  with  one  arm  about  ten  inches  long,  the  other  forty.  The 
bend  in  the  tube  is  filled  with  mercury,  and  water  is  poured  into 
the  long  arm.  We  then  have  a  long  column  of  water  sustaining  a 
short  column  of  mercury,  the  heights  being  inversely  as  the  densi- 
ties. By  the  side  of  this  tube  is  a  barometer,  made  by  closing  a 
common  glass  tube  at  one  end,  filling  with  mercury,  and  inverting 
over  a  cistern  containing  the  same  liquid.  The  precautions  and 
details  will  be  found  under  Experiment  No.  55.  By  the  side  of 
this  tube  is  placed  a  rod  about  ten  inches  long,  sharply  pointed  at 
both  ends,  and  capable  of  moving  up  and  down  so  as  to  touch  the 
surface  of  the  mercury  in  the  barometer  cistern.  A  steel  scale 
divided  into  millimetres  is  adjacent  to  both  tubes,  so  that  it  can  be 
read  at  the  same  time  as  thg  mercury  columns. 

Experiment.  Focus  the  telescope  so  that  both  scale  and  mer- 
cury are  distinctly  visible. 
Then  raise  it  until  it  is 
nearly  on  a  level  with  A, 
the  top  of  the  column  of 
water,  and  bring  its  hori- 
zontal cross-hair  exactly  to 
coincide  by  the  slow  mo- 
tion. Read  the  scale,  di- 
viding the  millimetres  into 
tenths  by  the  eye.  Do  the 
same  at  JB  and  (7;  then  the 
difference  in  height  of  A 
I  ™*  i  and  (7,  divided  by  that  of 
C  and  B,  will  equal  the 
specific  gravity  of  the  mer- 
cury, which  should  be  compared  with  its  true  value.  As  the  sur- 
face of  mercury  is  curved  upwards,  that  of  water  downwards,  the 
cross-hairs  should  be  brought  to  the  top  of  the  former,  and  to  the 
bottom  of  the  latter.  If  great  accuracy  is  required  in  this  experi- 
ment, allow  for  the  meniscus,  or  curved  portion  at  the  top  of  the 


Fig.  11. 


HOOK    GAUGE.  41 

water,  by  adding  one  half  its  thickness  to  the  height  of  the  water 
column. 

Next  raise  the  rod  EF,  and  read  the  height,  first  of  the  top 
and  then  of  the  bottom.  The  difference  will  be  its  length. 
It  is  safer  to  test  the  result  by  moving  it  and  repeating.  Then 
bring  the  rod  so  that  it  shall  just  touch  the  surface  of  the  mercury, 
that  is,  so  that  the  point  and  its  reflection  shall  coincide,  and  read 
the  height  of  Z>,  and  of  the  top  of  the  rod.  Their  difference 
added  to  the  length  of  the  rod  gives  the  height  of  the  column. 
Read  the  height  of  the  standard  barometer  placed  among  the 
meteorological  instruments.  Reduce  this  to  millimetres,  and  sub- 
tract from  it  the  other  measurement.  The  difference  will  be  the 
depression  caused  by  air  and  the  other  errors  in  the  barometer  D. 

13.    HOOK  GAUGE. 

Apparatus.  A  stand,  Fig.  12,  on  which  may  be  placed  a  vessel 
of  water  A,  and  a  micrometer  screw  j5,  by  which  we  can  raise  or 
lower  a  rod  carrying  two  points,  one  turned  upwards,  the  other 
downwards. 

Experiment.  Fill  up  the  vessel  until  the  water  just  covers  the 
point  of  the  hook.  Then  turn  the  screw  so  that 
upon  looking  at  the  reflection  on  the  surface  of 
some  object  as  a  window  sash,  a  slight  distortion  is 
produced  by  the  elevation  of  the  water  above  the 
hook.  Make  ten  measurements,  moving  the 
screw  after  each,  take  their  mean  and  compute  the 
probable  error  of  a  single  observation.  When 
the  point  is  raised  it  draws  the  liquid  with  it. 
Screw  it  down  until  it  touches  the  liquid,  and  C 
read  the  micrometer,  then  raise  it  until  the  liquid  Flg<  12' 

separates,  and  take  ten  readings  in  each  position.  Compute,  as  be- 
fore, the  probable  error,  and  reduce  to  fractions  of  a  millimetre, 
which  is  easily  done  if  the  pitch  of  the  screw  is  known.  This 
gives  a  measure  of  the  comparative  accuracy  of  the  hook  and 
simple  point.  Both  are  used  for  determining  the  exact  height  of 
any  liquid  surface,  the  hook  being  employed  most  frequently  in 
this  country,  the  point  abroad.  When  the  surface  of  a  liquid  is 


42  SPHEROMETER. 

rising  or  falling,  and  we  wish  to  know  the  exact  time  when  it 
reaches  a  given  level,  we  should  use  the  hook  when  it  descends, 
otherwise  the  point ;  because  the  former  should  always  be  brought 
up  to  the  surface,  the  latter  down  to  it. 

This  instrument  is  so  extremely  delicate  that  it  will  show  the 
lowering  of  a  surface  of  water  in  a  few  minutes  by  evaporation. 
A  variety  of  interesting  researches  may  be  conducted  with  it,  by 
the  different  students  of  a  class.  Thus  its  comparative  accuracy 
with  water,  mercury  and  other  liquids,  may  be  measured,  their 
rate  of  evaporation,  and  the  effect  of  impurities,  such  as  a  drop  of 
oil.  The  height  to  which  a  liquid  may  be  raised  by  the  point,  is 
also  a  test  of  its  viscosity. 

14.  SPHEROMETER. 

Apparatus.  Two  lenses,  one  convex,  the  other  concave,  a  piece 
of  thick  plate  glass  and  a  spherometer.  The  latter  consists  of  a 
tripod,  with  a  micrometer  screw  in  the  centre,  whose  point  may  be 
moved  to  any  desired  distance  above  or  below  the  plane  of  the 
three  legs  on  which  it  rests.  The  most  important  qualities  are 
lightness  and  stiffness,  and  on  this  account  a  very  cheap,  and  quite 
efficient  spherometer  may  be  made  with  the  nut  and  tripod  of 
wood,  using  for  legs,  pieces  of  knitting  needles. 

Experiment.  Stand  the  spherometer  on  the  sheet  of  plate  glass 
and  turn  the  screw  until  its  point  is  in  contact  with  it.  There  are 
three  ways  of  determining  the  exact  position  of  contact.  The  first 
method  is  dependent  on  the  fact  that  if  the  point  of  the  screw  is 
too  low  the  spherometer  will  stand  unsteadily,  like  a  table  with 
one  leg  too  short.  The  screw  is  therefore  depressed  until  the  in- 
strument rattles,  when  its  top  is  moved  gently  from  side  to  side. 
An  exceedingly  small  motion  of  this  kind  is  perceptible  to  the 
hand.  The  screw  is  then  turned  up  and  down  until  the  exact 
point  of  contact  is  found.  The  second,  and  probably  the  best 
method,  is  to  turn  the  screw  slowly,  taking  care  that  no  greater 
pressure  is  exerted  on  one  leg  than  on  the  other ;  as  soon  as  the 
point  touches  the  glass  the  pressure  is  removed  from  the  legs,  and 
the  friction  of  the  nut  at  once  makes  the  whole  instrument  revolve. 
Care  must  be  taken  not  to  press  on  the  top  of  the  screw,  or  the 
tripod  will  be  bent,  and  an  incorrect  reading  obtained.  The 
third  method  of  determining  contact  depends  on  the  sound  pro- 


SPHEROMETER.  43 

duced  when  the  instrument  slides  over  the  glass,  which  changes 
when  the  screw  touches  the  surface.  It  should  be  moved  but  a 
short  distance  and  without  pressure,  for  fear  of  scratching  the  glass. 

Having  determined  this  point  with  accuracy,  read  the  position 
of  the  screw,  taking  the  number  of  revolutions  from  the  index  on 
one  side,  and  the  fraction  from  the  divided  circle. 

Place  the  spherometer  on  each  face  of  the  two  lenses  and  meas- 
ure the  position  of  the  point  of  contact  as  before.  Of  course  the 
screw  must  be  raised  when  the  surface  is  convex,  and  depressed 
when  it  is  concave.  Subtract  each  of  these  readings  from  that 
taken  on  the  plate  glass,  and  the  difference  gives  the  height  of  a 
segment  of  the  sphere  to  be  measured,  whose  base  is  a  circle  pass- 
ing through  the  three  feet  of  the  spherometer.  Call  this  height  h 
the  radius  of  the  circle  r,  and  the  radius  of 
the  sphere  H ;  then  we  have,  Fig.  13,  AB 
=  h,  BD  =  r,  and  A  C  —  E.  But  by  sim- 
ilar triangles  AB  :  DB  =  DB  :  BE,  or 

r2          h 
h  :  r  =  r  :  2  J2  —  A,  or  H  =  ^r  +  -5-     Com- 

—ii         A 

pute  in  this  way  the  radius  of  each  surface 
of  the  lenses,  remembering  that  a  negative  r.    r 

radius  denotes  a  concave  surface.  To  de- 
termine r,  measure  the  distance  of  each  leg  of  the  spherometer 
from  the  axis  of  the  screw,  and  take  their  mean.  Measure  also  the 
distances  of  the  three  legs  from  each  other  and  take  their  mean. 
They  form  the  three  sides  of  an  equilateral  triangle ;  compute  by 
geometry  the  radius  of  the  circumscribed  circle,  and  see  if  this 
value  of  r  agrees  with  that  previously  found.  Both  r  and  h  must 
be  taken  in  the  same  unit,  as  millimetres  or  inches,  and  great  care 
should  be  taken  to  make  no  mistake  in  the  position  of  the  decimal 
point.  The  reduction  of  h  is  effected  by  multiplying  it  by  the  pitch 
of  the  screw. 

Finally,  compute  the  principal  focal  distance,  F,  by  the  formula 

-^  =  (n  —  1)  I  -^  -j-   -vp  ,  in  which  H  and  Rr  are  the  radii  of 

the  two  surfaces,  as  computed  above,  and  n  the  index  of  refraction 
of  the  glass.  The  latter  varies  in  different  specimens,  but  in  com- 
mon lenses  is  about  1.53. 


44  RATING  CHRONOMETERS. 

15.    ESTIMATION  OP  TENTHS  OP  A  SECOND. 

Apparatus.  A  heavy  body  carrying  a  small  vertical  mirror  is 
suspended  by  a  wire,  so  that  it  will  swing  by  torsion,  about  once 
in  half  a  minute.  A  small  telescope  with  cross  hairs  in  its  eye- 
piece, is  pointed  towards  the  mirror,  and  a  plate  with  a  pin  hole  in 
it  is  placed  in  such  a  position  that  when  the  mirror  swings,  the 
image  of  the  hole  will  pass  slowly  across  the  field  of  view  of  the 
telescope,  like  a  star.  It  may  be  made  bright  by  placing  a  mirror 
behind  it  and  reflecting  the  light  of  the  window.  The  whole  ap- 
paratus should  be  enclosed  so  as  to  cut  off  stray  light.  A  good 
clock  beating  seconds  is  also  needed. 

Experiment.  Twist  the  mirror  slightly,  so  that  it  shall  turn 
slowly.  On  looking  through  the  telescope  a  point  of  light  or  star 
will  be  seen  to  cross  the  field  of  view,  at  equal  intervals  of  about 
half  a  minute.  Note  the  hour  and  minute,  and  as  the  star  ap- 
proaches the  vertical  line  take  the  seconds  from  the  clock  and 
count  the  ticks  of  the  pendulum.  Fix  the  eye  on  the  star  and 
note  its  position  the  second  before,  and  that  after,  it  passes  the 
wire.  Subdividing  the  interval  by  the  eye  we  may  estimate  the 
true  time  of  transit  within  a  tenth  of  a  second.  Take  twenty  or 
thirty  such  observations  and  write  them  in  a  column,  and  in  a  sec- 
ond column  give  their  first  differences.  '  Take  their  mean  and 
compute  the  probable  error.  It  will  show  how  accurately  you  can 
estimate  these  fractions  of  seconds. 

This  is  called  the  eye  and  ear  method  of  taking  transits,  which 
form  the  basis  of  our  knowledge  of  almost  all  the  motions  of  the 
heavenly  bodies.  It  is  still  much  used  abroad,  although  in  this 
country  superseded  in  a  great  measure  by  the  electric  chronograph 
described  on  p.  16. 

16.     RATING  CHRONOMETERS. 

Apparatus.  Two  timekeepers  giving  seconds,  one,  which  may 
be  the  laboratory  clock,  to  be  taken  as  a  standard,  and  a  second  to 
be  compared  with  it.  For  the  latter  a  cheap  watch  may  be  kept 
expressly  for  the  purpose,  or  the  student  may  use  his  own.  If  the 
true  time  is  also  to  be  obtained,  a  transit  or  sextant  is  needed  in 
addition. 

Experiment.  First,  to  obtain  the  true  time.  As  this  problem 
belongs  to  astronomy  rather  than  physics,  a  brief  description  only 


RATING    CHRONOMETERS.  45 

will  be  given.  It  may  be  done  in  two  ways,  with  a  transit  or  a 
sextant ;  the  former  being  used  in  astronomical  observations,  the 
latter  at  sea.  A  transit  is  a  telescope,  mounted  so  that  it  will 
move  only  in  the  meridian.  With  it  note  by  the  clock  the 
minute  and  second  when  the  eastern  and  western  edges  of  the  sun 
cross  its  vertical  wire,  and  take  their  mean.  Correct  this  by  the 
amount  that  the  sun  is  slow  or  fast,  as  given  in  the  Nautical  Al- 
manac, and  we  have  the  instant  of  true  noon.  The  interval  be- 
tween this  and  twelve,  as  given  by  the  clock,  is  the  error  of  the 
latter. 

The  sextant  may  be  used  at  any  time  when  the  sun  is  not  too 
near  either  the  meridian  or  the  horizon.  A  vessel  containing  mer- 
cury is  used,  called  an  artificial  horizon,  and  the  distance  between 
the  sun  and  its  image  in  this  is  measured.  Since  the  surface  of  the 
mercury  is  perfectly  horizontal,  this  distance  evidently  equals  ex- 
actly twice  the  sun's  altitude.  If  the  observation  is  made  in  the 
morning,  when  the  sun  is  ascending,  the  sextant  is  set  at  somewhat 
too  great  an  angle,  if  after  noon  at  too  small  an  angle,  and  the 
precise  instant  when  the  two  images  touch  is  noted  by  the  clock. 
The  sun's  altitude,  after  allowing  for  its  diameter,  is  thus  obtained. 
We  then  have  a  spherical  triangle,  formed  by  the  zenith  Z,  the 
pole  jP,  and  the  sun  &.  In  this,  PZ  is  given,  being  the  comple- 
ment of  the  latitude ;  PS,  the  sun's  north  polar  distance,  is  ob- 
tained from  the  Nautical  Almanac,  and  ZS  is  the  complement  of 
the  altitude  just  measured.  From  these  data  we  can  compute  the 
angle  ZPS,  which  corrected  as  before  and  reduced  to  hours,  min- 
utes and  seconds,  gives  the  time  before  or  after  noon.  The  practi- 
cal directions  for  doing  this  will  be  found  given  in  full  in  J?ow- 
ditches  Navigator. 

By  these  methods  we  obtain  the  mean  solar  time,  which  is  that 
used  in  every  day  life.  For  astronomical  purposes  sidereal  time, 
or  that  given  by  the  apparent  motion  of  the  stars,  is  preferable. 
It  is  found  by  similar  methods,  using  a  star  instead  of  the  sun. 

In  an  astronomical  observatory  it  is  found  best  not  to  attempt  to 
make  the  clock  keep  perfect  time,  but  only  to  make  sure  that  its 
rate,  or  the  amount  it  gains  or  loses  per  day,  shall  be  as  nearly  as 
possible  constant.  We  can  then  compute  the  error  E  at  any  given 
time  very  easily  by  the  formula  E=  Ef  +  tr,  in  which  E'  was  the 


46  .     MAKING    WEIGHTS. 

error  t  days  ago,  and  r  the  rate.  By  transposing  we  may  also  ob- 
tain r,  when  we  know  the  errors  E  and  E',  at  two  times  separated 
by  an  interval  t.  Take  the  last,  two  observations  of  the  clock- 
error,  which  should  be  recorded  in  a  book  kept  for  the  purpose, 
and  compute  the  error  at  the  time  of  your  observation,  and  see 
how  it  agrees  with  your  measurement. 

If  the  day  is  cloudy,  or  no  instruments  are  provided  for  determ- 
ining the  true  time,  the  experiment  may  be  performed  as  follows. 
Compute,  as  above,  the  rate  and  error  of  the  clock.  Next  take 
the  difference  in  minutes  and  seconds  between  the  clock  and  the 
watch  to  be  compared.  To  obtain  the  exact  interval,  a  few  sec- 
onds before  the  beginning  of  the  minute  by  the  watch,  note  the 
time  given  by  the  clock,  and  begin  counting  seconds  by  the  ticks 
of  the  pendulum.  Then  fixing  your  eyes  on  the  watch,  mark  the 
number  counted  when  the  seconds'  hand  is  at  zero.  Repeat  two 
or  three  times,  until  you  get  the  interval  within  a  single  second. 
Now  correcting  this  by  the  error  of  the  clock,  taking  care  to  give 
the  proper  signs,  we  get  the  error  of  the  watch.  The  next  thing 
is  to  set  the  watch  so  that  it  shall  be  correct  within  a  second.  For 
this  purpose  it  must  be  stopped,  by  opening  it  and  touching  the 
rim  of  the  balance  wheel  very  carefully  with  a  piece  of  paper,  or 
other  similar  object.  Set  the  minute  hand  a  few  minutes  ahead  to 
allow  for  the  following  computation.  Subtract  the  clock-error  from 
the  time  now  given  by  the  watch.  It  will  give  the  time  by  the 
clock,  at  which  if  the  watch  is  started  it  will  be  exactly  right.  A 
few  seconds  before  this  time  hold  the  watch  horizontally,  with  the 
fingers  around  the  rim,  and  at  the  precise  second  turn  to  the  right 
and  then  back.  The  impulse  starts  the  balance-wheel,  and  the 
watch  will  now  go,  differing  from  the  clock  by  an  amount  just 
equal  to  the  error  of  the  latter. 

17.    MAKING  WEIGHTS. 

Apparatus.  A  very  delicate  balance  and  set  of  weights,  some 
sheet  metal,  a  pair  of  scissors,  a  millimetre  scale,  and  a  small  piece 
of  brass,  A,  weighing  about  18.4  grammes.  The  weights  are  best 
made  of  platinum  and  aluminium  foil ;  but  where  expense  is  a 
consideration,  sheet  brass  may  be  used  for  the  heavier,  and  tin  foil 
for  the  lighter  weights.  To  improve  the  appearance  of  the  brass 
and  prevent  its  rusting  it  may  be  tinned,  or  dipped  in  a  silvering 


PROPER    METHOD    OF    WEIGHING.  47 

solution,  or  perhaps  better  still,  coated  with  nickel.  Some  steel 
punches  for  marking  the  numbers  0,  1,  2  and  5,  a  mallet  and  sheet 
of  lead  should  also  be  provided. 

PROPER  METHOD  OF  WEIGHING. 

A  good  balance  is  so  delicate  an  instrument  that  the  utmost  care 
is  needed  in  using  it.  The  student  should  thoroughly  understand 
its  principle,  and  know  how  to  test)  both  its  accuracy  and  delicacy. 
See  Measurement  of  Weights,  p.  19.  The  beam  should  never  be 
left  resting  on  its  knife-edges,  or  they  will  become  dulled.  It  is 
therefore  commonly  made  so  that  it  may  be  lifted  off  of  them  by 
turning  a  milled  head  in  front  of  the  balance.  A  second  milled 
head  is  also  added  to  raise  supports  under  each  scale-pan.  To 
weigh  any  object  the  following  plan  must  be  pursued.  "  To  see  if 
the  balance  is  in  good  order,  lower  the  supports  under  the  scale- 
pans,  then  those  under  the  beam,  by  turning  the  two  milled  heads. 
The  long  pointer  attached  to  the  beam  should  now  swing  very 
slowly  from  side  to  side,  and  finally  come  to  rest  at  the  zero.  Re- 
place the  supports,  and  open  the  glass  case  which  protects  the  bal- 
ance from  currents  of  air.  The  object  to  be  weighed,  if  metallic 
and  perfectly  dry,  may  be  placed  directly  on  the  scale-pan,  other- 
wise it  should  be  weighed  in  a  watch-glass  whose  weight  is  after- 
wards determined  separately.  Now  place  one  of  the  weights  in 
the  opposite  scale-pan,  and  remove  the  supports  first  from  the  pans 
and  then  from  the  beam.  This  must  be  done  very  slowly  and 
carefully.  Students  are  liable  to  let  the  beam  fall  with  a  jerk 
on  the  knife-edges,  by  which  the  latter  are  soon  dulled  and  ruined. 
An  accurate  weighing  is  necessarily  a  slow  process  and  should 
never  be  attempted  when  one  is  in  a  hurry.  Moreover,  by  re- 
moving the  supports  quickly  the  scale-pans  are  set  swinging,  and 
the  beam  itself  vibrating  through  a  large  arc,  so  that  it  will 
not  come  to  rest  for  a  long  time.  It  is  better  while  using  the 
larger  weights  to  lower  the  supports  a  very  small  amount  only, 
and  notice  which  way  the  index  moves.  As  it  is  below  the  beam 
it  always  moves  towards  the  lighter  side.  The  smaller  weights 
must  be  touched  only  with  a  pair  of  forceps,  as  the  moisture  of  the 
fingers  would  soon  rust  them.  Those  over  100  grms.  may  be  taken 
up  in  the  hand  by  the  knob,  but  no  other  part  of  them  should  be 


48  PROPER    METHOD    OF    WEIGHING. 

touched.  Weights  should  never  be  laid  down  except  on  the  scale- 
pans,  or  in  their  places  in  the  box.  Now  try  weighing  the  piece 
of  brass  A.  Lay  it  on  one  scale-pan,  and  a  10  gr.  weight  on  the 
opposite  side.  The  index  moves  towards  the  latter  when  the  sup- 
ports are  removed,  as  described  above.  Replace  the  10  grs.  by 
20  grs.  This  is  too  heavy,  and  the  index  moves  the  other  way. 
Try  the  10  grs.  and  5  grs. -too  light;  add  2  grs. -still  too  light; 
another  2  grs. -too  heavy;  replace  the  latter  by  1  gr.  —  too  light. 
The  weight  evidently  lies  between  18  and  19  grammes.  Add  the 
.5  gr.,  or  500  mgr. -too  heavy;  substitute  the  200  mgr.-too 
light,  and  so  go  on,  always  following  the  rule  of  taking  the  weights 
in  the  order  of  their  sizes,  and  never  adding  small  weights  by 
guess,  or  much  time  will  be  lost.  Having  determined  the  weight 
within  .01  gr.,  the  milligrammes  are  most  easily  found  by  a  rider. 
This  consists  of  a  small  wire  whose  weight  is  just  10  mgr.  It  is 
placed  on  different  parts  of  the  beam,  which  is  divided  like  a  steel- 
yard into  ten  equal  parts,  which  represent  milligrammes.  Thus  if 
the  rider  is  placed  at  the  point  marked  6,  or  at  a  distance  of  .6 
the  length  of  one  arm  of  the  balance,  it  produces  the  same  effect 
as  if  6  mgrs.  were  placed  in  the  scale-pan.  It  is  generally  arranged 
so  that  it  can  be  moved  along  the  beam  without  opening  the  glass 
case,  which  protects  the  latter  from  dust  and  currents  of  air.  By 
taking  care  to  lower  the  supports  of  the  beam  slowly,  as  recom- 
mended above,  the  swing  of  the  index  is  made  very  small ;  it  is 
sufficient  to  see  if  it  moves  an  equal  distance  on  each  side  of  the 
zero,  instead  of  waiting  for  it  to  come  absolutely  to  rest.  To  make 
sure  that  no  errror  is  made  in  coimting  the  weights,  their  sum 
should  be  taken  as  they  lie  in  the  scale-pan,  and  also  from  their 
vacant  places  in  the  box. 

Decimal  weights  are  made  in  the  ratio  of  1 ,  2  and  5,  and  their 
multiples  by  10,  and  its  powers.  To  obtain  the  4  and  the  9  it  is 
necessary  to  duplicate  either  the  1  or  the  2.  The  English  adopt 
the  former  method,  the  French  the  latter.  Comparing  the  two 
mathematically,  we  find  that  using  the  weights  5,  2,  2,  1,  we  shall, 
on  an  average  in  ten  weighings,  remove  a  weight  from  box  to 
scale-pan  34  times,  of  which  it  will  be  put  back  17  times  during 
the  weighing,  and  the  remaining  17  times  after  the  weighing  is 
completed.  In  the  English  method,  with  the  weights  5,  2,  1,  1, 


PROPER    METHOD    OF   WEIGHING.  49 

under  the  same  circumstances  the  weights  are  again  used  34  times, 
replaced  15  times  during  the  weighing,  and  19  after  it.  There  is 
therefore  no  difference  in  rapidity  of  one  plan  over  the  other.  The 
French  system  has,  however,  the  great  advantage  that  we  may  at 
any  time  test  our  weights  against  one  another,  since  1  -j-  2  -f-  2 
should  equal  the  5  weight,  and  sometimes  in  weighing,  if  a  mis- 
take  is  suspected  a  test  may  be  applied  by  using  the  additional 
weight  instead  of  putting  back  all  the  small  weights,  and  adding  a 
larger  one,  as  is  necessary  in  the  English  system.  To  meet  this 
difficulty  a  third  1  gramme  weight  is  sometimes  added  by  English 
makers. 

Experiment.  To  make  a  set  of  weights  for  weighing  fractions 
of  a  gramme.  Four  are  needed  of  platinum  or  brass  weighing 
500,  200,  200  and  100  mgrs.,  and  four  of  aluminum,  or  thick  tin 
foil,  weighing  50,  20,  20  and  10  mgrs.  The  latter  should  be  made 
first,  since  being  the  lightest  they  are  the  easiest  to  adjust.  Cut  a 
rectangle  of  the  foil  about  3  or  4  centimetres  on  a  side,  and  weigh 
it  within  a  milligramme.  Now  determine  its  area  by  measuring  its 
four  sides  and  taking  the  product  of  its  length  by  its  breadth.  If 
the  opposite  sides  are  not  equal,  take  their  mean.  Let  A  equal 
the  area,  and  TFthe  weight  of  the  foil.  Evidently  W  -f-  A  will 
equal  t0,  the  weight  per  square  millimetre,  and  50,  20  and  10  di- 
vided \)jw  will  give  the  areas  of  the  required  weights.  Cut  pieces 
somewhat  too  large  and  reduce  them  to  the  proper  size  by  the 
Method  of  Successive  Corrections,  p.  10.  This  is  accomplished  by 
weighing  each  and  dividing  its  excess  by  w.  The  quotient  shows 
how  much  should  be  cut  off.  As  they  cannot  easily  be  enlarged 
if  made  too  small,  and  the  thickness  of  the  foil  may  not  be  the 
same  throughout,  pieces  should  be  cut  off  smaller  than  the  com- 
puted excess.  Small  amounts  may  be  taken  from  the  corners,  and 
when  completed  one  of  the  latter  should  be  turned  up  to  make-  it 
easier  to  pick  them  up  with  the  forceps.  Finally,  lay  them  on  the 
plate  of  lead,  and  stamp  their  weight  in  milligrammes  on  them,  with 
the  steel  punches  and  mallet.  Do  the  same  with  the  heavier  foil, 
thus  making  the  500,  200,  200  and  100  mgrs.  weight.  More  care 
is  needed  with  them,  and  the  last  part  of  the  reduction  should  be 
effected  with  a  file.  Unless  great  care  is  taken,  two  or  three  will 


50 


DECANTING    GASES. 


be  spoiled  by  making  them  too  light,  before   one  of  the    right 
weight  is  obtained. 

18.     DECANTING  GASES. 

Apparatus.  A  pneumatic  trough,  which  is  best  made  of  wood 
lined  with  lead,  and  painted  over  with  paraffine  varnish.  A  gradu- 
ated glass  tube  Z>,  Fig.  14,  closed  at  one  end,  and  holding  about 
100  cm.3,  a  tubulated  bell-glass  B  containing  about  a  litre,  with 
stop-cock  G  attached,  and  two  or  three  dry  Florence  flasks.  The 
mouths  of  the  latter  should  be  ground,  so  that  they  may  be  closed 
by  a  plate  of  ground  glass ;  to  remove  the  moisture  they  should  be 
heated  in  a  large  sand  bath,  or  over  steam  pipes.  A  thermometer 
is  also  needed. 

Experiment.  Measure  the  temperature  of  the  air  in  the  flask  A 
by  the  thermometer,  also  its  moisture,  or  rather  its  dew-point.  The 

latter  may  be  assumed  to 
be  the  same  as  that  of  the 
room,  and  obtained  from 
the  student,  using  the  me- 
teorological instruments. 
Now  close  the  flask  with 
the  plate  of  glass,  and  im- 
merse it  neck  downwards 
in  the  pneumatic  trough. 
It  may  be  kept  in  this  po- 
sition for  any  length  of  time,  as  the  water  prevents  the  air  from 
escaping.  Next  fill  the  large  graduated  vessel  J3  with  water,  by 
opening  its  stop-cock  C  and  immersing,  then  close  C  and  raise 
it.  Now  decant  the  gas  into  it  by  pouring,  just  as  you  would  pour 
water,  only  that  it  ascends  instead  of  falling.  When  all  has  been 
transferred  read  very  carefully  the  volume,  as  given  by  the  gradu- 
ation, also  the  approximate  height  of  the  water  inside  above  that 
outside  the  jar.  Dividing  this  difference  by  13.6  the  specific  grav- 
ity of  mercury,  and  subtracting  the  quotient  from  the  height  of  the 
barometer,  gives  the  pressure- to  which  the  enclosed  air  is  subjected. 
Its  temperature  may  be  assumed  equal  to  that  of  the  water,  and  it 
may  be  regarded  as  saturated  with  moisture.  Next,  to  transfer  it 
into  the  graduated  tube  7>,  attach  a  rubber  tube  to  (7,  and  after  fill- 


REDUCTION    OF    GASES.  51 

ing  D  with  water  and  inverting  it  in  the  trough,  let  the  air  bubble 
into  it  from  the  tube  by  opening  G  and  lowering  JB.  When  D  is 
nearly  full,  close  (7  so  as  to  prevent  the  escape  of  the  air,  and  read 
and  record  the  volume  as  given  by  the  graduation  on  D.  Now 
decant  the  air  from  J)  into  the  flask  A.  Great  care  is  necessary  in 
this  operation  to  prevent  spilling,  and  it  is  best  to  practise  a  few 
times  beforehand,  until  it  can  be  transferred  without  allowing  a 
single  bubble  to  escape.  Continue  to  empty  B  until  all  the  air  has 
been  passed  into  A.  The  latter  will  then  be  nearly  full  of  air, 
unless  some  has  been  lost.  In  the  latter  case  do  not  give  up  the 
experiment,  but  keep  on,  retaining  as  much  air  as  possible.  Now 
holding  the  neck  of  the  flask  in  the  hand  press  the  ground  glass 
against  it  with  the  thumb,  so  as  to  retain  what  water  is  still  in  it, 
and  taking  it  out  of  the  trough  stand  it  on  the  table  right  side  up. 
Wipe  the  outside  dry,  and  weigh  it  in  its  present  condition ;  also 
when  full  of  water,  and  when  empty.  Call  the  three  weights  m, 
n  and  o,  respectively.  The  volume  of  air  in  cm.8,  at  the  beginning 
of  the  experiment,  will  equal  n  —  o  in  grammes ;  that  at  the  end 
n  —  m.  There  are  now  four  volumes  of  air  to  be  compared.  First 
the  volume  at  the  beginning  of  the  experiment,  when  the  air  was 
moist  and  the  dew-point  was  given  ;  secondly,  when  transferred 
to  £ ;  thirdly,  that  found  by  adding  the  readings  of  D ;  and 
fourthly,  that  at  the  close  of  the  experiment.  Reduce  all  of  these 
to  the  standard  pressure  and  temperature  by  the  method  given 
below,  when  they  should  be  equal  if  no  air  has  escaped,  other- 
wise the  difference  shows  the  amount  of  the  loss.  Great  accuracy 
must  not  be  expected,  owing  to  the  absorption  of  the  air  by  the 
water,  and  for  various  other  reasons. 

REDUCTION  OF  GASES  TO  STANDARD   TEMPERATURE  AND 
PRESSURE. 

I.  Dry  Gas.  Given  a  volume  T^P  of  dry  gas  at  temperature  £, 
and  barometric  pressure  P,  to  find  what  would  be  its  volume  T^H 
if  cooled  to  0°  C,  and  the  pressure  altered  to  the  standard  H  = 
760  m.m.  Suppose  that  it  is  first  cooled  to  0°,  without  changing 
the  pressure,  and  call  its  new  volume  T^P.  We  have  by  Gay 
Lussac's  law  for  the  expansion  of  gases,  FJP  =  T^P  (1  +  «£),  m 
which  a  =  2Tj,  the  coefficient  of  expansion  of  gas.  Again,  by 


52  STANDARDS    OF    VOLUME. 

Mariotte'e  law  we  have,  VOF  :  Vm  =  JET  :  P.      Hence  Vof  = 

JET  H 

VoH'->  or  substituting,  VtP  =  Ym-(l  +  «0*J  or» 


97.Q    p 

C\  ^ 


_ 

tp  (273  +  <)  760 

For  any  other  temperature  £',  and  pressure  JP',  we  have, 
273  Pr  T_    P   273  +  H 

FOH  =  Y«*  (273  +  0  760  '  hence  F*^  =  Fflp  P7'  273  +«'  ' 

The  first  formula  is  used  to  determine  the  true  quantity  of  gas 
present,  that  is,  the  volume  at  the  standard  temperature  and  pres- 
sure. The  second,  to  compute  the  new  volume  when  we  alter 
both  temperature  and  pressure. 

II.  Gas  saturated  with  Moisture.  Call  p  the  pressure  of 
aqueous  vapor  at  the  temperature  t.  Then  of  the  total  pressure  P 
we  have  p  due  to  the  vapor,  and  P  —  p  to  the  gas  ;  substitute? 
therefore,  P  —  p  for  P  in  equation  (1),  and  we  have, 

-^) 


III.  Gas  moist,  but  not  saturated.  Let  the  gas  be  gradually 
cooled,  until  the  temperature  becomes  so  low  that  the  moisture 
can  no  longer  be  retained  as  vapor,  but  begins  to  condense  on  the 
walls  of  the  vessel.  This  temperature  T  is  called  the  dew-point; 
let  p'  be  the  corresponding  pressure  of  the  vapor.  Then  p  :  p'  = 
1  +at  :  1  +  aT,  or  p  =  p'(\  +  at)  -T-  (1  +  aT),  and  substitut- 
ing this  value  in  equation  (3^),  we  have 

273  r  (1  +  at)  n 

y°R  ~    Ktp   (273  +  t)  760  '  Lr       "  p  (1  +  a  T)J  ' 

19.      STANDARDS  OF  VOLUME. 

Apparatus.  A  balance  AB,  Fig.  15,  capable  of  sustaining  5 
kgrs.  on  each  side,  and  turning  with  a  tenth  of  a  gramme  under 
this  load.  Remarkably  good  results  may  be  obtained  with  com- 
mon balances,  such  as  are  used  for  commercial  purposes,  by  attach- 
ing a  long  index  to  the  beam,  as  in  the  figure.  Several  pounds  of 
distilled  water  should  be  provided,  a  thermometer,  a  set  of  weights, 
and  a  rubber  tube  and  funnel.  Instead  of  a  scale-pan,  a  counter- 
poise C  is  attached  to  one  arm  of  the  balance  as  a  method  of 
double  weighing  is  to  be  used.  The  standard  to  be  graduated, 
which  we  will  suppose  to  be  a  tenth  of  a  cubic  foot,  consists  of  a 
glass  vessel  D,  whose  capacity  somewhat  exceeds  this  amount.  A 


STANDARDS    OF    VOLUME. 


53 


J"  steam  valv.e  is  screwed  into  the  cap  closing  the  lower  end 
which  also  carries  a  sharp  brass  point  to  form  the  lower  limit  of 
the  volume.  A  ring  is  attached  to  the  cap  closing  the  upper  end 
of  the  vessel,  by  which  the  whole  is  supported.  A  brass  hook 
with  the  point  turned  upwards  passes  through  this  cap,  in  which 
a  hole  has  been  drilled  to  allow  the  air  to  pass  in  or  out.  The 
hook  may  be  raised  or  lowered,  and  clamped  at  any  height  by  a 
conical  nut  surrounding  it,  or  by  a  set  screw.  Finally  a  millimetre 
scale  should  be  attached  to  the  upper  end  of  D. 

Experiment.    Note  the  height  of  the  barometer,  the  temperature 
of  the  room,  also  that  of  the  distilled  water.    Fill  D,  by  attaching 
the  rubber  tube,  as  in 
the  figure,  opening  E 
and    pouring    in    the 
water.    When  the  ves- 
sel   is    full,    close    E 
and  remove  the  rubber 
tube.     Take  care  that 
no  air  bubbles  adhere 
to    the    side    of    the      H 
glass.    •  Open   E   and 


Fig.  15. 


draw  off  the  water  until  it  stands   just  on  a 

level  with   the  top  of  the  scale   attached  to 

the  glass.     Counterpoise  by  adding  weights  to 

the  scale-pan  F,  until  the  index  stands  at  zero, 

first  reading  the  directions  for  weighing,  given 

on  page  47.    t)raw  off  enough  water  to  lower 

its    level    just    one    centimetre,    counterpoise 

again,  and  repeat  until  the  surface  reaches  the 

bottom  of  the  scale.    If  too  much  water  is  removed  at  any  time 

refill  the  vessel  above  the  mark,  and  draw  off  the  water  again. 

Now  bring  the  water  level  just  above  the  point  of  the  hook,  and 

close  E,  so  that  the  flow  shall  take  place  drop  by  drop.     Use  the 

hook  as   in  Experiment   13,  and   as  soon   as  the  point  becomes 

visible  close  E.     Read  the  level  of  the  water  and  counterpoise  as 

before.     Repeat  two  or  three  times,  adding  a  little  water  after  each 

measurement.    Now  open  JE7,  and  let  the  water  run  out  until  the 

lower  point  just  touches  the  surface.     Measure  the  temperature  of 

the  water  as  it  escapes.     To  counterpoise  the  beam  nearly  three 


54  STANDARDS    OF    VOLUME. 

kilogrammes  additional  must  be  added  to  F.  Make  this  weigh- 
ing with  care,  and  repeat  two  or  three  times,  as  when  observing 
the  upper  point.  Subtract  each  of  the  weights  when  the  vessel 
was  full,  from  the  mean  of  those  last  taken,  and  the  difference 
gives  the  weight  of  the  water  contained  between  the  lower  point 
and  each  of  the  other  observed  levels. 

Now  to  determine  the  volume,  we  have  given  by  Kater,  the 
weight  of  1  cubic  inch  of  distilled  water  at  62°  F.,  and  30  inches 
pressure,  equals  252.456  grains,  and  1  gramme  equals  15.432  grains. 
From  this  compute  the  weight  of  one  tenth  of  a  cubic  foot.  Two 
corrections  must  now  be  applied,  the  first  for  temperature,  the 
second  for  pressure.  Water  has  an  expansion  of  about  .00009 
per  1°  F.  when  near  62°,  and  glass  .000008  linear,  or  three  times 
this  amount  of  cubical  expansion  at  the  same  temperature;  of 
course  the  apparent  change  of  volume  is  the  difference  of  in- 
crease of  the  water,  and  of  the  glass.  Evidently  at  a  high 
temperature  less  water  would  be  required,  hence  this  correction  is 
negative  if  the  temperature  is  above  62°.  Practically  in  making 
standards  it  is  best  to  keep  the  temperature  exactly  at  62°,  adding 
ice  or  warm  water  if  necessary,  as  this  correction  is  a  little  doubt- 
ful, owing  to  the  unequal  expansion  of  different  specimens  of  glass. 
The  vessel  D  is  buoyed  up  by  the  air,  by  an  amount  equal  to  the 
weight  displaced,  and  this  weight  is  evidently  proportional  to  the 
barometric  pressure  H.  Now  100  cubic  inches  of  air  at  30  inches 
weigh  2.1  grms.,  hence  at  1  inch  it  would  be  f^,  and  if  the  pres- 
sure is  changed  from  30  to  H,  the  change  in  weight  would  evi- 
dently be  2.1  X  (30  —  H)  4-  30.  The  weights,  however,  are  also 
buoyed  up  in  the  same  way,  but  as  the  specific  gravity  of  brass 
is  about  8,  the  effect  is  only  one-eighth  as  great.  The  true 
correction  is  then  seven-eighths  of  this  amount.  The  higher  the 
barometer  the  greater  the  buoyancy,  and  the  lighter  the  water 
will  appear,  or  this  correction  will  be  negative  for  pressures  above 
30  inches.  Both  the  corrections  will  be  small,  and  in  most  cases 
can  be  neglected  ;  but  it  is  well  to  make  them,  in  order  to  be  sure 
to  understand  the  principle.  Having  thus  computed  how  much 
the  tenth  of  a  cubic  foot  ought  to  weigh,  see  if  the  distance  be- 
tween the  points  is  correct,  and  if  not,  determine  by  interpolation 


BEADING    MICROSCOPES.  55 

where  the  water  level  should  be  in  order  to  render  the  capacity 
exact. 

20.    READING  MICROSCOPES. 

Apparatus.  Three  cheap  French  microscopes  mounted  on 
moveable  stands,  as  in  AB,  Fig.  16.  Two  should  have  cross-hairs 
in  their  eye-pieces,  while  the  third  should  contain  a  thin  plate  of 
glass  with  a  very  fine  scale  ruled  on  it.  An  accurate  scale  divided 
into  millimetres  is  required  as  a  standard  of  comparison,  and  since 
the  division  marks  of  those  in  common  use  are  too  broad  for  exact 
measurements,  it  is  better  to  have  one  made  to  order,  with  very 
fine  lines  cut  on  the  centre  of  one  face  instead  of  on  the  edge. 
The  best  material  is  glass,  but  copper  or  steel  will  do,  especially  if 
coated  with  nickel  or  silver.  Several  objects  to  be  measured 
should  be  selected,  as  a  rod  pointed  at  each  end,  the  two  needle 
points  of  a  beam-compass,  and  a  scale  divided  into  tenths  of  an 
inch,  whose  correctness  is  to  be  tested.  Under  the  microscopes  is 
placed  a  board  D,  on  which  the  object  to  be  measured  (7,  is  laid, 
and  which  may  be  raised  or  lowered  gradually  by  screws,  or  fold- 
ing wedges.  Another  method  of  supporting  the  microscopes, 
superior  in  some  respects,  will  be  found  described  under  the  Ex- 
periment of  Dilatation  of  Solids  by  Heat. 

Experiment.  If  a  measurement  within  a  tenth  of  a  millimetre 
is  sufficiently  exact,  use  the  two  microscopes  with  cross-hairs. 
Place  them  at  such  a  distance  apart  that  each 
shall  be  over  the  end-  of  the  object  to  be  meas- 
ured, which  should  be  laid  on  D.  They 
should  be  raised  or  lowered  until  in  focus,  and 
then  set  so  that  their  cross-hairs  shall  exactly 
coincide  with  the  two  given  points.  Remove  Fis- 16- 

the  object  very  carefully,  so  as  not  to  disturb  their  position,  and 
replace  it  by  the  standard  scale,  bringing  the  zero  to  coincide 
with  one  of  the  cross-hairs.  Now  looking  through  the  other 
microscope  read  the  position  of  its  cross-hairs  on  the  scale,  esti- 
mating the  fractions  of  a  millimetre  in  tenths.  If  the  image  of 
the  scale  is  not  distinct  it  may  be  focussed  by  slowly  raising  or 
lowering  the  board  on  which  it  is  placed,  taking  great  care  not 
to  disturb  the  microscopes.  To  get  the  whole  number  of  milli- 
metres, a  needle  may  be  laid  down  on  the  scale,  and  the  right 
division  distinguished  by  its  point. 

If  greater  accuracy  is  desired,  use  the  third  microscope,  find- 


56  DIVIDING    ENGINE. 

ing  the  magnitude  of  the  divisions  of  its  scale  in  the  following 
manner;  focus  it  on  the  steel  scale,  placing  it  so  that  two  divi- 
sions of  the  latter  shall  be  in  the  field  at  the  same  time.  Read 
each  of  them  by  the  scale  in  the  eye-piece,  and  take  the  differ- 
ence ;  the  reciprocal  is  the  magnitude  of  one  division  in  millimetres. 
Repeat  a  number  of  times  and  take  the  mean.  To  make  any 
measurement,  place  this  microscope  with  one  of  the  others  over  the 
points  to  be  determined,  and  take  the  reading  with  its  scale,  esti- 
mating tenths  of  a  division ;  then  substitute  the  steel  scale  as  be- 
fore, and  read  the  millimetre  mark  preceding,  also  that  following. 
By  a  simple  interpolation  the  distance  is  obtained  from  these  three 
readings  with  great  accuracy. 

Try  both  these  methods  with  the  objects  to  be  measured,  and 
then  test  the  scale  of  tenths  of  an  inch  by  measuring  the  distance 
of  each  inch  mark  from  the  zero,  and  reducing  the  millimetres  to 
inches.  Measure  also  in  the  same  way  the  ten  divisions  of  one  of 
the  inches. 

One  of  the  best  ways  to  measure  off  a  large  distance,  as  ten  or 
twenty  metres,  with  accuracy,  is  by  means  of  a  couple  of  reading 
microscopes.  A  steel  rule  is  used,  the  ends  being  marked  by  the 
microscopes,  as  they  are  in  rough  measurements,  by  the  finger.  In 
all  cases  where  the  graduation  extends  to  the  end  of  the  rule  it  is 
better  to  use  the  mark  next  to  it,  both  as  being  more  accurate,  and 
as  affording  a  better  object  to  focus  on. 

21.    DIVIDING  ENGINE. 

Apparatus.  This  instrument  rests  on  a  substantial  stand 
ABED,  Fig.  17,  like  the  bed-plate  of  a  lathe.  A  carefully  con- 
structed micrometer  screw  moves  in  this,  and  pushes  a  nut  y  from 
end  to  end.  The  screw  should  have  a  pitch  of  about  a  millimetre, 
or  a  twentieth  of  an  inch,  if  English  measures  are  preferred.  The 
head  of  the  screw  is  divided  into  one  hundred  parts,  and  turns 
past  an  index  which  is  again  divided  into  ten  parts,  as  in  Fig.  4, 
]).  24.  The  screw  may  be  turned  by  a  milled  head  or  a  crank. 
The  nut  must  have,  a  bearing  of  considerable  length,  a  decimetre 
is  scarcely  too  much,  as  any  irregularities  are  thus  compensated. 
It  should  be  split  so  that  it  may  be  tightened  by  screws,  or  better, 
by  a  spring,  and  slides  along  two  guides,  AB  formed  like  an  in- 
verted V,  and  DE,  which  is  flat.  A  scale  is  cut  on  the  latter  to 
give  the  whole  number  of  revolutions  of  the  screw.  The  nut 


DIVIDING    ENGINE.  57 

should  move  with  perfect  smoothness  from  end  to  end,  but  not  too 
freely.  A  certain  amount  of  bac7c-lash  is  unavoidable  (that  is,  the 
screw  may  always  be  turned  a  short  distance  backwards  or  for- 
wards without  moving  the  nut),  but  this  does  no  harm,  as  when 
in  use  it  should  always  be  moved  in  the  same  direction.  A  second 
screw  similar  to  the  other,  but  smaller,  and  at  right  angles  to  it,  is 
attached  to  C,  so  that  its  nut  may  be  moved  backwards  or  for- 
wards about  one  decimetre.  It  carries  a  reading  microscope  J?, 
made  of  a  piece  of  light  brass  tubing,  by  inserting  an  eye-piece 
above,  and  screwing  a  microscope  objective  into  the  lower  end. 
It  may  be  focussed  by  sliding  the  tube  up  and  down  by  a  rack 
and  pinion.  Cross-hairs  should  be  placed  in  the  eye-piece,  but 
in  some  cases  a  fine  scale,  or  eye-piece  micrometer,  is  preferable. 

To  use  this  instrument  as  a  dividing  engine,  the  microscope  must 
be  made  movable,  so  that  it  can  be  replaced  by  a  graver  for  metals, 
or  a  pen  for  paper.  The  micrometer  head  F  has  ten  equidistant 
holes  cut  in  it,  in  which  steel  pins  can  be  inserted.  These  strike 
against  a  stop  which  they  cannot  pass  unless  it  is  pushed  down  by 
the  finger.  A  sheet  of  thick  plate  glass  DSTE  serves  as  a  stand 
on  which  to  lay  objects,  and  under  it  is  a  large  mirror  to  illuminate 
them,  but  it  may  be  removed  when  desired. 

Experiment.  This  instrument  may  be  applied  to  a  great  variety 
of  purposes.  Several  experiments  with  it  will  therefore  be  de- 
scribed. 

1st.  To  test  the  screw.  Lay  a  glass  plate  divided  into  tenths 
of  a  millimetre  on  DSTE, 
and  bring  the  microscope 
over  it.  Use  a  moderately 
high  power,  as  a  J"  objective, 
and  focus  on  the  scale;  the 
want  of  a  fine  adjustment 
may  be  partly  remedied  by 
varying  the  distance  of  the 
eye-piece  from  the  objective. 
Bring  the  first  division  of  the 
scale  to  coincide  with  the 
cross-hairs  of  the  microscope 
by  turning  the  micrometer- 
head  F.  Read  the  whole  number  of  turns  from  the  scale  on  DE, 
and  the  fraction  from  F.  Move  it  one  or  two  turns  to  the  right, 
and  set  again ;  repeat  several  times,  and  compute  the  probable  error 


58  DIVIDING    ENGINE. 

of  one  observation.  It  equals  the  error  of  setting.  Turn  the 
screw  the  other  way,  and  bring  it  back  to  the  line.  The  differ- 
ence between  this  reading  or  the  mean  of  ten  such  readings,  and 
that  previously  obtained,  gives  the  back-lash.  Set  in  turn  on  sev- 
eral successive  points  of  the  scale.  The  first  differences  should  be 
equal.  Mark  two  crosses  on  a  plate  of  glass  with  a  diamond,  three 
or  four  centimetres  apart.  Measure  the  interval  between  them 
with  different  portions  of  the  screw,  and  see  if  they  agree.  If  not, 
the  defect  in  the  screw  must  be  carefully  examined,  and  corrections 
computed.  The  screw  M should  be  similarly  tested. 

2d.  Determination  of  the  pitch  of  the  screw.  Procure  a 
standard  decimetre  (or  other  measure  of  length)  and  measure  the 
distance  between  its  ends.  The  temperature  should  be  nearly  that 
taken  as  a  standard,  or  if  great  accuracy  is  required,  allowance  made 
for  the  difference  of  expansion  of  the  screw  and  decimetre.  From 
this  measurement,  which  should  be  repeated  several  times,  compute 
the  true  pitch  of  the  screw,  and  the  correction  which  must  be  ap- 
plied when  distances  are  measured  with  it. 

3d.  To  measure  any  distance.  Lny  the  object  on  the  glass 
plate  and  bring  the  cross-hairs  of  the  microscope  to  coincide  first 
with  one  end  of  it,  and  then  with  the  other.  The  difference  in 
the  readings  is  the  length.  Apply  to  it  the  correction  previously 
determined. 

4th.  To  determine  the  form  of  any  curved  line.  For  exam- 
ple, use  one  of  the  curves  drawn  by  a  tuning  fork,  in  the  Experi- 
ment on  Acoustic  Curves.  Bring  the  cross-hairs  to  coincide  with 
several  points  in  turn  of  one  of  the  sinuosities,  and  read  both 
micrometer  heads.  These  give  two  coordinates,  from  which  the 
points  of  the  curve  may  be  constructed  on  a  large  scale,  and  com- 
pared with  the  curve  of  sines,  the  form  given  by  theory.  The 
relative  positions  of  a  number  of  detached  points  may  also  be  thus 
determined,  as  in  the  photographs  of  the  Pleiades  and  other 
groups  of  stars  by  Mr.  Rutherford. 

5th.  Graduation.  For  a  first  attempt,  make  a  scale  on  paper 
with  a  pencil  or  pen.  Replace  the  microscope  by  a  hard  pencil 
with  a  flat,  but  very  sharp  point.  It  must  be  arranged  so  that  it 
can  be  moved  backwards  or  forwards  a  limited  distance,  but  not 
sideways.  Every  fifth  line  should  be  longer  than  the  rest,  which 


RULING    SCALES.  59 

should  be  exactly  equal  to  each  other  in  length.  Fasten  the  paper 
securely  on  the  glass  plate  so  that  it  shall  not  slip.  Suppose  now 
lines  are  to  be  drawn  at  intervals  of  half  a  millimetre.  Insert  a 
pin  in  one  of  the  holes  in  F,  and  turn  the  latter  to  the  stop. 
Draw  a  line  with  the  pencil  for  the  beginning  of  the  scale,  depress 
the  stop  to  let  the  pin  pass,  give  F  one  turn,  bring  the  pin  again 
to  the  stop  and  draw  a  second  line,  and  so  on.  If  the  lines  are  to 
be  a  millimetre  apart,  draw  one  line  for  every  two  turns.  In  the 
same  way,  by  inserting  more  pins  a  finer  graduation  may  be  ob- 
tained. Instead  of  using  the  pins  a  table  may  be  computed 
beforehand,  giving  the  reading  of  the  screw  for  each  line  to  be 
drawn,  allowing  for  the  errors  of  the  screw,  if  great  accuracy  is  re- 
quired. The  scale  is  then  ruled  by  bringing  the  nut  successively 
into  the  various  positions  marked  in  the  table,  and  drawing  a  line 
after  each. 

A  most  important  application  of  this  instrument  is  to  the  meas- 
urement of  photographs  of  the  sun  taken  during  eclipses.  The 
position  of  the  moon  at  any  instant  is  thus  obtained,  with  a  degree 
of  precision  otherwise  unattainable.  In  this,  and  other  cases 
where  angles  must  also  be  measured,  the  plate  of  glass  ES  should 
be  removed,  and  the  object  laid  on  a  rotary  stand,  with  a  gradu- 
ation showing  the  angle  through  which  it  is  turned. 

22.    RULING  SCALES. 

Apparatus.  In  Fig.  18,  two  strips  of  wood  A  and  J?,  rest  on  a 
smooth  board,  and  are  held  in  place  by  the  weights  C  and  D. 
The  ends  of  a  string  are  attached  to  them,  which  is  stretched  by 
means  of  a  weight  F,  so  that  if  C  and  D  are  raised  A  and  J5  will 
slide.  A  peg  is  inserted  in  B,  which  moves  between  two  steel 
plates  fastened  to  A,  one  being  fixed,  the  other  movable  by  means 
of  a  screw  Gr.  If,  then,  either  weight  is  raised,  the  strip  of  wood 
on  which  it  rests  will  be  drawn  forward  by  F,  but  will  be  free  to 
move  through  a  space  equal  to  the  difference  of  the  diameter  of 
the  peg  and  the  interval  between  the  two  steel  plates.  If  desired, 
Gr  may  be  a  micrometer  screw,  by  which  this  interval  may  always 
be  accurately  determined.  It  may  be  fastened  in  any  position  by 
a  clamp  or  set  screw.  A  steel  rod  H  is  used  to  draw  the  division 
lines.  It  is  fixed  at  one  end,  and  carries  at  the  other  a  pencil,  pen, 
graver  or  diamond,  according  as  the  lines  are  to  be  drawn  on  pa- 
per, metal  or  glass.  By  this  arrangement  there  is  little  or  no 


60  RULING    SCALES. 

lateral  motion  of  the  graver,  but  unfortunately  it  draws  a  curved 
line.  To  remedy  this  defect,  the  rod  may  be  replaced  by  a  stretched 
wire,  to  the  centre  of  which  the  graver  is  attached,  or  the  latter 
may  slide  past  a  guide  against  which  it  is  pressed  by  a  spring. 

Experiment.  For  many  purposes  in  using  a  scale,  it  makes  but 
little  difference  what  the  divisions  are,  provided  that  they  are  all 
equal,  and  this  is  especially  the  case  in  all  accurate  measurements, 
since  as  a  correction  must  always  be  made  for  temperature,  we 
can  readily  at  the  same  time  correct  for  the  size  of  the  divisions. 
The  instrument  here  described  will  probably  give  divisions  more 
nearly  equal  than  those  obtained  by  a  micrometer  screw,  but  it  is 
more  difficult  to  make  them  of  any  exact  magnitude,  since  any 
deviation  is  multiplied  by  the  number  of  divisions. 

To  draw  a  scale,  lay  a  piece  of  paper  on  B  and  fasten  it  with 
tacks  or  clips.  To  secure  uniformity  in  the  length  of  the  long  and 

short  division  marks,  rule  three 
parallel  lines  as  limits,  attach 
a  sharp  flat-pointed  pencil  to' 
H,  and  slide  A  and  B  until 
the  beginning  of  the  scale 
is  under  JET.  Draw  a  line  with 
the  latter,  and  make  one  stroke 
with  the  machine.  This  is  done 
by  raising  (7,  when  F  will  draw 

A  forward  a  distance  equal  to  the  interval  between  the  two  plates 
near  Gr,  minus  the  thickness  of  the  peg.  Lay  C  down  and  raise 
D.  A  will  now  remain  at  rest,  but  B  will  move  through  the  same 
distance.  Draw  a  second  line  with  the  pencil,  and  repeat,  making 
every  fifth  line  about  twice  as  long  as  the  others.  They  will  be 
found  spaced  at  distances  which  may  be  regulated  by  the  screw  Gr. 
Try  making  short  scales  in  the  same  way,  with  large  and  small 
divisions.  It  is  always  safer  to  keep  the  hand  on  one  weight  while 
the  other  is  lifted.  The  magnitude  of  F  should  be  such  that  the 
strajn  on  the  cord  will  be  greater  than  the  friction  of  repose  when 
the  weights  are  up,  but  less  than  the  friction  of  motion  when  they 
are  down.  If  F  is  too  light,  when  C  is  raised  A  will  not  start ; 
if  too  heavy,  it  will  strike  so  hard  that  it  will  move  B.  To  test 
the  accuracy  of  the  machine  draw  a  single  line,  take  a  hundred 


RULING    SCALES.  61 

strokes  and  draw  another.  Then  without  moving  G-  push  the 
slides  back  and  draw  a  third  line  close  to  the  first ;  take  a  hundred 
strokes  and  draw  a  fourth  line  near  the  second.  Measure  the  in- 
terval between  the  first  and  third,  and  the  second  and  fourth. 
They  should  be  equal,  but  if  not,  the  difference  divided  by  an 
hundred  gives  the  average  difference  in  length  of  a  stroke  the 
second  time,  compared  with  the  first. 

Instead  of  a  pencil,  a  pen  may  be  used  to  draw  the  lines,  or  a 
graver,  if  a  metallic  scale  is  desired.  The  finest  scales  are  ruled 
on  glass  by  a  diamond.  Instead  of  using  the  natural  edge  of  the 
gem,  as  when  cutting  glass,  an  engraver's  diamond  should  be  em- 
ployed, which  is  ground  with  a  conical  point ;  the  direction  in 
which  it  should  be  held,  and  the  proper  pressure,  being  obtained  by 
trial.  Scales  may  also  be  etched  by  covering  the  surface  with  a 
thin  coating  of  wax  or  varnish,  and  the  lines  marked  with  a 
graver.  If  metallic,  it  is  then  subjected  to  the  action  of  nitric 
acid ;  if  of  glass,  to  the  fumes  of  fluorhydric  acid.  It  is  possible 
that  the  new  method  of  cutting  glass  by  a  sand-blast  may  prove 
applicable  to  this  purpose  with  a  great  saving  of  time  and  trouble. 


MECHANICS  OF  SOLIDS. 


23.     COMPOSITION  OF  FORCES. 

Apparatus.  Two  pulleys  A  and  .Z?,  Fig.  19,  are  attached  to  a 
board  which  is  hung  vertically  against  a  wall.  Two  threads  pass 
orer  them,  and  a  third  (7,  is  fastened  to  their  ends  at  D.  Three 
forces  may  now  be  applied  by  attaching  weights  to  the  ends  of 
the  cords.  The  weights  of  an  Atwood's  machine  are  of  a  con- 
venient form,  but  links  of  a  chain,  picture  hooks,  cents,  or  any 
objects  of  nearly  equal  weight  may  be  used.  Small  beads  are 
attached  to  the  three  threads  at  distances  of  just  a  decimetre 
from  D. 

Experiment.     Attach  weights  2,  3  and  4  to  the  three  cords,  and 
let  D  assume  its  position  of  equilibrium.     Owing  to  friction  it  will 
remain  at  rest  in  various  neighboring  positions, 
their  centre  being  the  true  one.     Now  meas- 
ure the  distance  of  each  bead  from  the  other 
two  witn  a   millimetre  scale,   and    obtain  -the 
angle   directly   from  a   table    of   chords.      If 
these  are  not  at  hand,  dividing  the  distance 
by  two,  gives  the  natural  sine  of  one  half  the 
required  angle.     By  the  law  of  the  parallelo- 
gram of  forces,  the  latter  are  proportional  to 
the  sides  of  a  triangle  having  the  directions  of 
the  forces.     But  these  sides  are  proportional  to  the  sines  of  the 
opposite  angles,  hence  the  sines  of  the  angles  included  between  the 
threads  should  be  proportional  to  the  forces  or  weights  applied. 
Divide  the  two  larger  forces  by  the  smaller,  and  do  the  same  with 
the  sines  of  the  angles,  and  see  if  the  ratios  are  the  same.      The 
angles  themselves  should  first  be  tested  by  taking  their  sum,  which 
should   equal  360°.     If  either  angle  is  nearly  180°,  it  cannot  be 
accurately  measured  in  this  way,  but  must  be  found  by  subtracting 
the  sum  of  the  other  two  from  360°,  or  measuring  one  side  from 


Fig.  19. 


MOMENTS.  63 

the  prolongation  of  the  other.  It  is  well  to  draw  the  forces  from 
the  measurement,  and  see  if  a  geometrical  construction  gives  the 
same  result  as  that  obtained  by  calculation.  Repeat  with  forces  in 
several  other  ratios,  as  3,  4,  5  ;  2,  2,  3 ;  3,  5,  7 ;  taking  care  in  all 
cases  to  include  in  the  weights  the  supports  on  which  they  rest. 

24.     MOMENTS. 

Apparatus.  A  board  AB,  Fig.  20,  is  supported  at  its  centre  of 
gravity  on  the  pin  O.  It  should  revolve  freely,  and  come  to  rest 
in  all  positions  equally.  Two  forces  may  be  applied  to  it  by 
the  weights  D  and  E,  attached  by  threads  to  the  pins  F  and  6r. 
Their  magnitudes  may  be  varied  from  1  to  10  by  different  weights, 
and  their  points  of  application  by  using  different  pins,  as  H,  I  and 
J.  To  measure  their  perpendicular  distances  from  the  pin  C,  a 
wooden  right-angled  triangle  or  square  is  provided,  one  edge  of 
which  is  divided  into  millimetres,  or  tenths  of  an  inch. 

^Experiment.     Various  laws  of  forces  may  be  provred  with  this 
apparatus.    1st.  When  a  single  force  acts  on  a  body  AB  fixed  at  one 
point,  as  (7,  there  will  be  equilibrium  only  when  it  passes  through 
this  point.     Remove  FD  and  attach  a  weight  E  to  &.    It  will  be 
found  that  the  body  will  remain  at  rest  only  when  the  point  G  is 
in  line  with  M  and  C.     2d.  A  force  produces  the  same  effect  if 
applied  at  any  point  along  the  line  in  which  it  tends  to  move  the 
body.    Apply  the  two  weights  D  and  E,  which  tend  to  turn  the 
board  in  opposite  directions.     Make 
their  ratio  such  that  MGr  shall  be  in 
line  with  G-,  H,  J.    Now  transfer  the 
end  of  the  thread  from  G-  to  Jf,  ./and 
<7in  turn,  when  it  will  be  found  that 
the  position  of  the  board  will  be  un- 
changed.   It  should  be  noticed,  how- 
rig.  20. 
ever,  that  in  the  last  case  the  board  is 

in  unstable  equilibrium,  since  FJ  falls  beyond  the  point  of  support 
C.  3d.  The  moment  of  a  force,  or  its  tendency  to  make  a  body 
revolve,  is  proportional  to  the  product  of  its  magnitude  by  its  per- 
pendicular distance  from  the  point  of  support.  Make  D  equal  2, 
and  attach  it  to  JT,  so  that  the  thread  rests  over  the  edge  of  the 
board,  which  is  the  arc  of  a  circle  with  centre  at  (7,  and  radius  .6. 
Its  tendency  to  make  the  board  revolve  is  therefore  the  same,  what- 


64 


PARALLEL    FORCES. 


ever  the  position  of  the  latter.  Make  E  successively  1,  2,  3, 4,  5, 
6,  and  measure  the  perpendicular  distance  of  the  thread  to  which 
it  is  attached  in  each  case  from  C.  This  distance  is  measured  by 
resting  the  triangle  against  the  thread  and  measuring  the  distance 
of  6rby  its  graduated  edge.  In  each  case  the  moment  of  E  will  be 
found  to  be  the  same,  and  equal  to  2  X  6,  the  moment  of  D.  4th. 
When  two  forces  hold  the  body  in  equilibrium  their  resultant  must 
pass  through  the  fixed  point.  Make  D  equal  2,  and  attach  it  to 
F,  and  E  equal  3,  applied  at  G.  Lay  a  sheet  of  paper  on  the  right 
hand  portion  of  AB,  making  holes  for  F,  C  and  J~to  pass.  Draw 
on  it  with  a  ruler  the  direction  of  the  two  threads  prolonged,  and 
then  removing  it,  construct  their  resultant  geometrically  by  means 
of  the  parallelogram  of  forces.  It  will  be  found  to  pass  through 
C.  Repeat  two  or  three  times  with  different  weights  and  points 
of  application. 

25.    PARALLEL  FORCES. 

Apparatus.  The  apparatus  used  is  shown  in  Fig.  21.  AJB  is  a 
straight  rod  about  two  feet  long,  with  a  paper  scale  divided  into 
tenths  of  an  inch  attached  to  it.  It  is  supported  by  a  scale-beam 
CD  with  a  counterpoise,  so  that  it  is  freely  balanced,  and  remains 
horizontal.  Weights  formed  like  those  of  a  platform  scale  may  be 
attached  to  it  at  any  point,  by  riders,  as  at  E,  F  and  O.  Taking 
each  rider  as  unity,  four  sets  of  weights  are  required  of  magni- 
tudes 10,  5,  2,  2,  1,  .5,  .2,  .2,  .1.  Two  other  beams,  like  CD, 
should  also  be  provided,  to  which  these  weights  may  be  attached, 
as  at  E,  so  as  to  produce  an  upward  force  of  any  desired  magni- 
tude. All  these  scale-beams  may  be  very  roughly  made,  even 
a  piece  of  wood  supported  at  the  centre  by  a  cord,  being  suf- 
ficiently accurate.  English  beams  of  iron  may,  however,  be  ob- 
tained at  a  very  low  price. 


The 


Fig.  21. 


resultant  of  any  system  of  parallel  forces 
lying  in  one  plane  may  be  found  by 
this  apparatus.  Thus  suppose  we 
have  a  force  of  15.7  acting  upwards, 
and  two  of  8.3  and  1.4  acting  down- 
wards, and  distant  from  the  first 
6.4  and  8.7  inches  respectively. 
Produce  the  upward  force  by  add- 
ing the  weights  14.7  to  E,  and  the 


PARALLEL    FORCES.  65 

two  downward  forces  by  weights  7.3  and  .4  (allowing  1  for  each 
of  the  scale-pans)  at  F  and  G,  setting  them  at  the  points  of  the 
beam  marked  3.6  and  18.7.  They  are  then  at  the  proper  distance 
from  C,  which  is  at  10  inches  from  the  end.  We  now  find  that 
A  goes  down  and  B  up ;  by  placing  the  finger  on  the  beam  we 
see  that  it  can  be  balanced  only  by  applying  a  downward  force  to 
the  right  of  C.  Now  place  a  rider  in  this  position,  and  move  it 
backwards  and  forwards,  varying  the  weight  on  it  until  the  beam 
is  exactly  balanced.  The  magnitude  of  this  weight  will  be  found 
to  be  6,  and  its  position  16.8,  or  6.8  inches  from  C.  The  resultant 
of  the  three  forces  will  be  just  equal  and  opposite  to  this.  Had 
the  force  required  to  balance  them  acted  upwards,  we  should  havo 
used  one  of  the  auxiliary  scale-beams.  To  test  the  correctness  of 
this  result  we  compute  the  resultant  thus :  JR,  =  15.7  —  (8.3  -f- 
1.4)  =  6,  and  taking  moments  around  C  we  have  8.3  X  6.4  —  1.4 
X  8.7  =  6  X  aj,  or  x  =  6.8  the  observed  distances. 

Determine  the  position  and  magnitude  of  the  resultant  in  sev- 
eral similar  cases,  as  for  example  the  following,  in  which  IT  means 
an  upward,  and  D  an  downward  force,  and  each  is  followed  first  by 
its  magnitude,  and  then  by  the  point  on  the  bar  at  which  it  is  to 
be  applied. 

t     D,  5.0,  4.3  ;   Z7J  10.0,  10.0. 

2.  D,  '2.6,  3.2 ;   IT,  7.8,  10.0. 

3.  D,  7.4,  3.7 ;   IT,  17.1, 10.0. 

4.  D,  11.1,  2.1 ;  D,  6.5,  5.6;   IT,  2.3,  18.4. 

5.  D,  5.2,  1.9;   U,  15.2,  10.0  ;  D,  8.4,  12.6;   IT,  3.0,  18.1. 

Two  equal  parallel  forces  acting  in  opposite  directions  and  not 
in  the  same  line,  form  what  is  called  a  couple,  and  have  no  single 
resultant.  Thus  apply  the  two  forces  D,  12.0, 5.0,  and  IT,  12.0, 10.0. 
No  single  force  will  balance  the  beam.  Equilibrium  is  obtained 
only  by  a  second  couple  having  the  same  moment,  and  turning  in 
the  opposite  direction ;  thus  the  moment  being  12.0  X  5.0  =  60.0, 
the  beam  may  be  balanced  by  two  forces  of  10.0,  each  distant  6 
inches  from  one  another,  placing  the  upward  force  to  the  left. 
Find  in  the  same  way  some  equivalent  to  D,  4.3,  7.6,  and  U,  4.3, 
10.0,  and  notice  that  it  makes  no  difference  to  what  part  of  the 
beam  the  two  forces  are  applied,  provided  their  distance  apart 
remains  unchanged. 

5 


00  CENTRE    OF    GRAVITY. 

This  same  apparatus  may  be  applied  to  illustrate  the  case  of  a 
body  with  one  point  fixed,  acted  on  by  parallel  forces,  as,  for  ex- 
ample, the  lever,  by  using  a  stand  IT  with  two  pins,  between  which 
the  beam  may  turn.  This  stand  is  also  useful  in  finding  the  point 
of  application  of  the  resultant  in  the  above  cases. 

26.     CENTRE  OF  GRAVITY. 

Apparatus.  Several  four-sided  pieces  of  cardboard  (not  rec- 
tangles) and  a  plumb  line,  made  by  suspending  a  small  leaden 
weight  by  a  thread,  from  a  needle  with  sealing  wax  head. 

Experiment.  Make  four  holes  in  the  cardboard,  two  AB,  Fig. 
22,  close  to  two  adjacent  corners,  the  others  in  any  other  part  not 
too  near  the  centre.  Pass  the  needle  through  A  and  support  the 
cardboard  by  it ;  the  thread  will  hang  vertically  downwards,  and 
the  centre  of  gravity  must  lie  somewhere  in  this  line,  or  it  would 
not  be  in  equilibrium.  Mark  a  point  on  this  line  as  low  down  as 
possible,  and  connect  it  with  the  pin  hole.  Do  the  same  with  B ; 
the  intersection  of  the  two  is  the  centre  of  gravity.  Turn  the 
cardboard  over  and  repeat  with  the  other  holes.  This  gives  two 
determinations  of  the  centre  of  gravity.  To  see  if  the  two  points 
are  opposite  one  another,  prick  through  one  and  see  if  the  hole  coin- 
cides with  the  other.  By  suspending  at  any  other  points,  the  same 
result  should  be  obtained.  Be  careful  that  the  holes  are  large 
enough  to  enable  the  card  to  swing  freely. 

Next,  lay  the  card  down  on  your  note  book  and  mark  the  four 
points  A,  JB,  C,  .D.  Connecting  them  with  lines  gives  a  duplicate  of 
the  cardboard.     On  this  construct  the  centre  of 
gravity  geometrically.    Divide  into  two  trian- 
gles by  connecting  A  O.    Bisect  AD  in  E,  and 
CD  in  F.     The   centre  of  gravity  of  A  CD 
must  lie  in  AF,  also  in  CE^  hence  at  G.     Ob- 
tain Q-'  by  a  similar  construction  with  AB  C. 
The  centre  of  gravity  of  the  whole  figure  must 
lie  in  GG-'.    Make  a  second  construction  by 
connecting  BD,  making  the  triangles  ABD  and  B  CD ;  the  in- 
tersection of   QGr   and  its   corresponding  line  gives  the   centre 
of  gravity.    Lay  the  piece  of  cardboard  on  the  figure  and  prick 


CATENARY. 


67 


through  the  two  centres  of  gravity  previously  found.     They  should 
agree  closely  with  that  found  geometrically. 

27.     CATENARY. 

Apparatus.  A  chain  three  or  four  yards  long,  each  link  of  which 
is  a  sphere,  known  in  the  trade  as  a  ball  link  chain.  Every  tenth 
link  should  be  painted  black,  and  the  fiftieths  red.  A  horizontal 
scale  A B  (7,  Fig.  23,  attached  to  the  wall,  also  a  number  of  pins  to 
which  the  chain  may  be  fastened  by  short  wire  hooks,  and  its 
length  altered  at  will.  A  graduated  rod  BD  is  used  to  measure 
the  vertical  height  of  any  point  of  the  chain. 


_O— |-<-H— 1~- H-+-T 


Experiment.  First,  to  determine  the  average  length  of  the 
links.  Let  the  chain  hang  vertically  from  A,  measure  the  length 
of  each  hundred  links,  and  take 
their  mean.  A  simple  proportion 
gives  the  number  of  links  to 
which  AC  is  equal.  Suspend 
the  chain  at  A  and  (7,  making  the 
flexure  at  the  centre  about  half  a 
foot.  Measure  it  exactly,  and  in- 
crease the  original  length  10  links 
at  a  time  to  100.  Increase  it  also 
by  17  links,  by  63  and  by  48,  and 
measure  as  before.  Write  the  re- 


Fig.  23. 


suits  in  a  column  and  take  the  first,  second  and  third  differences 
of  the  first  measurements.  Now  obtain  by  interpolation  the  three 
values  for  17,  63  and  48  links,  and  compare  with  their  measured 
values. 

Next  suspend  the  chain  as  in  ADE,  and  measure  the  deflection 
at  intervals  of  five  inches  horizontally.  This  is  best  done  by  pass- 
ing a  pin  through  the  graduated  rod  at  the  zero  point,  letting  it 
hang  vertically,  then  measuring  by  it.  Taking  differences  as  be- 
fore, those  of  the  first  order  will  be  at  first  negative,  then  increase 
until  they  become  positive.  Where  the  first  difference  is  zero, 
is  evidently  the  lowest  point  of  the  curve.  By  the  method  of 
inverse  interpolation  find  this  point,  treating  the  first  differences 
as  if  they  were  the  original  variable,  and  recollecting  that  each 
difference  belongs  approximately  to  the  point  midway  between  the 


68  CRANK    MOTION. 

two  terms  from  which  it  was  obtained.  Thus  the  difference  ob- 
tained from  the  5  and  10  inches  corresponds  to  7£.  Obtain  this 
point  also  by  measurement,  by  laying  off  BF  equal  to  CJE,  pro- 
longing EF  to  Gr  and  measuring  GF.  A  C  minus  one-half  GE 
will  equal  the  required  distance.  Repeat  with  several  points 
below  E,  and  compare  with  the  computed  position  of  the  lowest 
point. 

28.     CRANK  MOTION. 

Apparatus.  A  steel  scale  AB,  Fig.  24,  divided  into  millime- 
tres, slides -in  a  groove  so  that  its  position  maybe  read  by  an  index 
E.  It  is  connected  by  the  rod  AD  to  the  arm  of  the  protractor, 
whose  centre  is  C.  On  turning  CD,  which  carries  a  vernier  F, 
AB  moves  backwards  and  forwards.  Several  holes  are  cut  in  AD 
so  that  its  length  may  be  altered  at  will. 

Experiment.  Make  AD  as  long  as  possible.  Measure  CD  by 
turning  it  until  D  is  in  line  with  C  and  A,  and  read  E\  then  turn  it 

180°,  and  read  again.  One-half  the 
difference  of  these  readings  equals 
CD.  Next,  to  find  the  reading  of 
the  vernier  when  CD  and  DA  are 
in  line.  Make  ACD  about  90° 
and  read  E  and  F.  Turn  CD  un- 
til the  reading  of  E  is  again  the 

same  and  read  F.  The  mean  of  these  two  readings  gives  the  re- 
quired point.  Repeat  two  or  three  times,  and  take  the  mean. 

Let  AS  represent  the  piston  rod  of  an  engine,  and  CD  the 
crank  attached  to  the  fly-wheel.  The  problem  is  to  determine  the 
relative  positions  of  these  two,  during  one  revolution.  Bring  D 
in  line  with  CA,  and  move  it  10°  at  a  time  through  one  revolu- 
tion, reading  E  in  each  case.  Do  the  same,  using  a  shorter  con- 
necting rod,  so  that  AD  shall  be  about  two  or  three  times  CD. 
To  compare  these  results  with  theory,  first  suppose  the  rod  CD 
infinitely  long.  The  distance  of  AB  from  the  mean  position  will 
then  always  equal  CD  X  cos  A  CD.  This  is  readily  computed 
from  the  accompanying  table  of  natural  cosines.  If,  as  is  most 
convenient,  CD  is  made  just  equal  to  1  decimetre,  the  distances  are 
given  directly  in  the  second  column  of  the  table  by  moving  the 


HOOK  S    UNIVERSAL   JOINT. 


69 


Angle. 

Cosine. 

0° 

1.000 

10° 

.985 

20° 

.937 

30° 

.866 

40° 

.766 

50° 

.643 

60° 

.500 

70° 

.342 

80° 

.174 

90° 

.000 

decimal  point  two  places  to  the  right.     Compare  these   results 

with   your  observations.     Construct  a  curve  in  which  abscissas 

represent' the  computed  positions  of  AJB,  and  or- 

dinates  the  difference  between  the  observed  and 

computed  results,  enlarging  the  scale  ten  times. 

If  a  smooth  curve  is  thus  obtained  it  is  probably 

due  to  the  short  length  of  AD.     The  correction 

due    to   this  is  readily  proved  to  be  AD    — 

*/AD2 —  CD2  sin2  A  CD,   or    calling    the   ratio 

AD  -^-CD—n,  it  is  AD  (n  —  J  n*  —  mtf'ACD). 

Compute  this  correction  for  every  30°,  knowing 

that  sin2  30°=  .25,  sin2  60°  =  .75.     The  points 

thus  obtained  should  lie  on  the  residual  curve  found  above.    Do 

the  same  with  the  shorter  arm  AD. 

29.     HOOK'S  UNIVERSAL  JOINT. 

Apparatus.  A  model  of  this  joint  with  graduated  circles  at- 
tached to  its  axles.  The  latter  should  be  so  connected  that  they 
may  be  set  at  any  angle. 

Experiment.  Set  the  axes  at  an  angle  of  45°,  and  bringing  one 
index  to  0°,  the  reading  of  the  other  will  be  the  same.  Now  move 
the  first  5°  at  a  time  to  180°,  and  read  the  other  in  each  position. 
Record  the  results  in  columns,  giving  in  the  first  the  reading  of 
one  index,  in  the  second  that  of  the  other,  and  in  the  third  their 
difference,  which  will  be  sometimes  positive,  and  sometimes  nega- 
tive. Construct  a  curve  with  abscissas  taken  from,  the  first  col- 
umn, and  ordinates  from  the  third,  enlarging  the  latter  ten  times. 
It  shows  how  much  one  wheel  gets  behind,  or  in  advance  of,  the 
other.  To  compare  this  result  with  theory,  let  Fig  25  represent  a 
plan  of  the  joint,  AC  and  CD  being  the  two 
axes.  Describe  a  sphere  with  their  intersec- 
tion C  as  a  centre.  The  great  circle  CD  is 
the  path  described  by  the  ends  of  one  hook, 
CE  that  described  by  the  other.  D  and  E 
must,  by  the  construction  of  the  apparatus, 
always  be  90°  apart.  Then  in  the  spherical 
triangle  CDUwe  have  given  J)E  —  90°,  ECD  =  45°,  the  angle 
between  the  axes,  and  one  side  as  CD,  and  we  wish  to  compute 


.25. 


70  COEFFICIENT    OF    FRICTION. 

CE.  But  by  spherical  trigonometry,  tang  CE  =  tang  CD  cos 
ECD.  Substituting  in  turn  CD  =  5°,  10°,  15°,  20°,  &c.,  we 
compute  the  corresponding  angle  through  which  the  second  wheel 
has  been  turned.  Construct  a  second  curve  on  the  same  sheet  as 
the  other,  using  the  same  scale.  Their  agreement  proves  the  cor- 
rectness of  both. 

Experiments  like  Nos.  28  and  29  may  be  multiplied  almost  in- 
definitely. Thus  various  forms  of  parallel  motion,  the  conversion 
of  rotary  into  rectilinear  motion  by  cams,  link  motion,  gearing, 
and,  in  fact,  almost  all  mechanical  devices  for  altering  the  path  of 
a  moving  body  may  be  tested  and  compared  with  theory. 

30.     COEFFICIENT  OF  FRICTION. 

Apparatus.  AJ3,  Fig.  26,  is  a  board  along  which  a  block  C  is 
drawn  by  a  cord  passing  over  a  pulley  J9,  and  stretched  by  weights 
placed  in  the  scale  pan  E.  The  friction  is  produced  between  the 
surfaces  of  C  and  A  J?,  which  should  be  made  so  that  they  may  be 
covered  with  thin  layers  of  various  substances  as  different  kinds 
of  wood,  iron,  brass,  glass,  leather,  <fcc.  C  is  made  of  such  a  shape 
that  by  turning  it  over  the  area  of  the  surface  in  contact  may  be 
altered.  The  pressure  on  C  and  the  tension  of  the  cord  may  also 
be  varied  at  will,  by  weights. 

Experiment.  Weights  are  added  to  E  in  regular  order,  as  when 
weighing,  and  the  tension  in  each  case  compared  with  the  friction 

of  C.  Friction  may  be  of  two 
kinds ;  first,  that  required  to  start  a 
body  at  rest,  called  the  friction  of 
repose,  and  secondly,  the  friction  of 
motion,  or  that  produced  when  the 
bodies  are  moving.  To  measure 
the  friction  of  repose,  see  if  the 
Fig- 26.  weight  is  capable  of  starting  the 

body  when  at  rest,  if  so,  stop  it  and  repeat,  varying  the  weight 
until  a  tension  is  obtained  sufficient  sometimes  to  start  it  and 
sometimes  not.  This  friction  is  very  irregular,  varying  with  dif- 
ferent parts  of  every  surface,  and  with  the  time  during  which  the 
two  substances  have  been  in  contact.  It  is  but  little  used  practi- 
cally, since  the  least  jar  converts  it  into  the  friction  of  motion. 
The  latter  is  much  less  than  the  friction  of  repose,  and  more  uni- 


k T 


B 


ANGLE    OF    FRICTION.  71 

form.  It  is  found  by  tapping  the  body  so  that  it  will  move,  and 
seeing  if  the  velocity  increases  or  diminishes.  In  the  first  case 
the  weight  in  J^is  too  large,  in  the  second  too  small. 

The  first  law  of  friction  is  that  the  friction  is  proportional  to 
the  pressure.  The  ratio  of  these  two  quantities  is  called  the  co- 
efficient of  friction.  Make  -the  load  on  (7,  including  its  own 
weight,  equal  to  1,  2,  3,  4,  5  kgs.  in  turn,  and  measure  the  friction. 
The  latter  equals  the  weight  of  E  plus  the  load  added  to  it  minus 
the  friction  of  the  pulley.  If  great  accuracy  is  required,  a  table 
should  be  prepared,  giving  the  magnitude  of  the  latter  for  differ- 
ent loads.  Compute  the  coefficient  of  friction  from  the  observa- 
tions, and  if  the  law  is  correct  they  should  all  give  the  same  re- 
sult. Measure,  in  each  case,  the  friction  of  repose  and  of  motion, 
and  notice  that  the  latter  is  always  much  the  smaller. 

Secondly,  the  friction  is  independent  of  the  extent  of  the  sur- 
faces in  contact.  This  law  follows  from  the  preceding,  but  it  is 
well  to  prove  it  independently  by  turning  C  on  its  different  sides, 
so  as  to  vary  the  areas  in  contact.  The  friction  will  be  found  to 
be  the  same  in  each  case.  Finally,  measure  the  coefficients  in  a 
number  of  cases,  and  compare  the  results  with  those  given  in  the 
tables  of  friction. 

31.    ANGLE  OF  FRICTION. 

Apparatus.  In  Fig.  27,  AB  is  a  stand  with  an  upright  B  C. 
AD  is  a  board  hinged  at  A,  which  may  be  set  at  any  angle  by  a 
cord  passing  over  the  pulley  C.  The  hinge  is  best  made  of  soft 
leather  held  by  a  strip  of  brass,  and  its  distance  from  the  upright 
should  be  just  one  metre.  A  scale  of  millimetres  is  attached  to 
the  upright,  and  a  wire  parallel  to  AD  serves  as  an  index.  Evi- 
dently the  reading  of  the  scale  gives  the  natural  tangent  of  the 
angle  of  inclination  DAB.  A  cord  attached  to  D  passes  over 
the  pulley  6",  around  the  wheel  F,  and  is  stretched  by  the  counter- 
poise 6r.  F  may  be  clamped  in  any  position,  or  turned  by  a  crank 
attached  to  it.  AD  may  thus  be  set  at  any  angle,  and  its  position 
is  to  be  determined  when  so  inclined  that  any  given  body,  as  E,  is 
just  on  the  point  of  sliding.  E  may  be  made  exactly  like  the 
sliding  mass  in  Experiment  30,  but  to  measure  its  friction  of  mo- 
tion a  fine  wire  should  be  attached  to  it  and  wound  around  the 
axle  of  F.  When  the  crank  is  turned  raising  AD,  the  body  E  is 
thus  drawn  slowly  down  the  inclined  plane.  In  order  that  it  may 
not  move  too  rapidly  this  portion  of  the  axle  should  be  much 


72 


BREAKING    WEIGHT. 


smaller  than  that  around  which  the  cord  OF  is  wound.  EF 
should  be  a  wire,  as  if  a  thread  is  used  it  will  stretch,  giving  E  an 
irregular  motion,  alternately  starting  and  stopping. 

Experiment.  To  measure  the  coefficient  of  friction  of  repose, 
turn  the  crank  until  the  body  begins  to  slide  ;  the  reading  of  the 

^  scale  gives  the  tangent  of 

the  angle  of  inclination. 
But  decomposing  the  weight 
of  E  into  two  parts,  paral- 
lel and  perpendicular  to  the 
plane,  their  ratio  will  equal 
the  coefficient  of  friction, 
and  also  the  tangent  of  the 
inclination.  Hence  the  co- 
efficient of  friction  is  given 
directly  by  the  scale.  Meas- 
ure again  the  coefficients  found  in  Experiment  30,  and  see  if  the 
results  agree  with  those  then  obtained. 

32.     BREAKING  WEIGHT. 

Apparatus.  In  Fig.  28,  _Z?  is  a  thumb  screw,  by  which  a  spring 
balance  A  may  be  drawn  back  so  as  to  exert  a  strain  on  the  wire 
CD.  Near  C  is  placed  a  spring  buffer  so  that  when  the  wire 
breaks,  the  jar  may  be  diminished.  D  may  be  a  simple  peg  to 
which  the  wire  is  attached,  or  a  spool  with  a  clamp  by  which  it  is 
held  at  any  point.  A  convenient  method  of  connection  is  to  at- 
tach one  end  of  the  wire  to  a  chain,  either  link  of  which  may  be 
passed  over  the  peg  J9,  according  to  the  length  employed.  To 
test  the  accuracy  of  the  balance  a  cord  may  be  substituted  for 
CD,  passed  over  a  pulley  E,  and  stretched  by  weights.  Some 
cord  and  fine  copper  or  iron  wire  of  various  sizes  should  be  fur- 
nished, also  a  gauge  to  measure  its  diameter. 

Experiment.  Fasten  a  cord  to  (7,  and  pulling  it  over  E,  record 
the  reading  of  A,  when  by  attaching  weights,  the  strain  is  made 
in  turn  0,  5,  10,  15,  20,  &c.,  pounds.  To  eliminate  the  friction  of 
the  pulley,  turn  7?  first  in  one  direction  and  then  in  the  other,  and 
take  the  mean  of  the  readings  of  A  in  each  case.  Now  construct 
a  residual  curve,  in  which  abscissas  represent  the  reading  of  A, 


2)    Jg 


LAWS    OF    TENSION.  73 

and  ordinates  the  difference  between  this  reading  and  the  weight 
applied.     From  this  curve  we  can  readily  determine  the  true  strain, 
knowing   the  reading  of  A,  how- 
ever inaccurate  the  latter  may  be. 

To  measure  the  breaking  weight 
of  any  body,  attach  one  end  of  it       . 
to  C  and  the  other  to  one  link  of         u 
the  chain.     Pass  the  latter  over  the 
peg  D,  so  that   C  shall  be  a  short 

distance  from  its  spring  buffer.  Turn  B  and  watch  the  index  of 
-4,  until  the  cord  breaks.  A  small  block  of  wood  may  be  placed 
in  front  of  the  index  to  show  the  greatest  tension  attained,  but 
care  must  be  taken  that  it  is  not  disturbed  by  the  recoil.  Repeat 
several  times  with  other  portions  of  the  same  cord  and  take  the 
mean  of  the  observed  maximum  tensions.  Do  the  same  with  some 
specimens  of  wire,  and  compute  their  tenacity  T,  or  strength  per 
square  inch.  For  this  purpose  measure  their  diameter  d  with  ac- 
curacy, by  the  gauge,  and  calling  TFthe  breaking  weight,  we  have, 


33.    LAWS  OP  TENSION. 

Apparatus.  In  Fig.  29,  A  is  a  cast-iron  bracket  firmly  fastened 
to  the  wall.  A  hook  is  attached  at_Z?,  and  from  it  the  scale  pan  E 
is  hung  by  the  wire  or  rod  to  be  tested.  Weights  may  then  be 
applied  so  as  to  give  any  desired  tension  to  the  latter.  A  brass 
rod  JBD  hangs  by  the  side  of  BC,  being  fastened  to  it  by  a  small 
clamp  at  the  top.  Fine  lines  are  drawn  on  both  rods,  and  their 
relative  change  in  position  measures  the  elongation  of  J3C.  Gr  is 
a  reading  microscope  made  of  a  brass  tube  about  6"  long,  with  the 
Microscopical  Society's  screw  cut  in  one  end,  so  that  any  micro- 
scope objective  may  be  used  with  it.  A  positive  eyepiece  with  a 
scale  at  its  focus  is  slipped  into  the  other  end.  This  microscope  is 
mounted  like  a  cathetometer,  by  fastening  it  to  a  vertical  brass 
tube  screwed  into  the  base  of  a  music  stand.  It  may  be  raised  or 
lowered,  and  held  at  any  point  by  a  set  screw.  The  following 
additional  apparatus  is  also  needed.  Several  wires  or  rods  of  va- 
rious materials,  as  wood,  copper,  brass,  iron,  lead,  and  some  of  the 
same  material  but  different  diameters.  A  millimetre  scale  to  meas- 
ure their  lengths,  and  a  Brown  &  Sharpe's  sheet  metal  gauge  to 
give  their  diameters.  Also  a  set  of  large  weights  to  vary  the  ten- 


74 


LAWS    OF    TENSION. 


a 


Fig.  29. 


sion.  To  prevent  too  sudden  a  jar  on  the  wire,  a  board  should  be 
placed  under  E  to  support  it  when  a  weight  is  added,  then  lower- 
ing it  by  means  of  a  screw. 

Experiment.  To  measure  the  extension  in  any  case,  attach  the 
wire  to  be  tried  to  the. hooks  at  IB  and  (7,  and  clamp  JBD  to  its 
upper  end.  Draw  a  line  on  J3  C  opposite 
one  of  those  on  J3D,  and  focus  the  micro- 
scope G-  on  them.  Read  their  relative  posi 
tion  by  means  of  the  scale  in  6r,  then  apply 
the  weight  and  read  again.  The  length  of 
J3C  is  thus  increased,  while  that  of  J3J)  is 
unaltered;  hence  the  change  in  their  relative 
distances  equals  the  extension.  To  reduce 
this  to  millimetres,  focus  the  microscope  on 
a  standard  millimetre,  and  thus  measure  the 
scale  in  G  directly.  The  distance  from  F  to 
the  clamp  is  measured  by  the  millimetre  scale, 
and  the  diameter  of  J3C\)j  the  gauge. 
The  laws  of  tension  may  now  be  determined. 
1st.  The  extension  is  proportional  to  the  length.  Use  a  copper 
wire  about  a  millimetre  in  diameter,  and  mark  on  it  a  number  of 
lines  at  different  heights.  Measure  the  extension  for  each  with  a 
load  of  20  kgs.  It  will  be  found  proportional  to  the  distances 
from  the  clamp.  2d.  The  extension  is  proportional  to  the  weight 
applied.  Measure  the  deflection  for  the  lower  lines,  increasing  the 
weight  2  kg.  at  a  time,  from  0  to  20  kilogrammes.  Removing  the 
latter  weight,  see  if  the  wire  has  returned  to  its  original  length  ; 
any  increase  is  called  the  permanent  set.  See  if  the  results  agree 
with  the  law.  3d.  The  elongations  are  inversely  proportional  to 
the  cross-section.  Try  the  series  of  wires  of  the  same  material, 
measuring  the  diameter  of  each  with  the  gauge,  and  using  the 
same  weight  for  all.  The  product  of  the  square  of  the  diameters 
by  the  elongation  should  be  constant.  The  modulus  of  elasticity 
is  the  force  which  would  be  required  to  double  the  length  of  a 
body  of  cross-section  unity,  supposing  this  could  be  done  without 
breaking  it,  or  changing  the  law  which  holds  for  small  weights. 
To  compute  it,  suppose  d  the  diameter,  I  the  length,  and  e  the  elon. 
gation  of  a  wire  under  a  tension  T.  If  the  cross-section  was 


CHANGE    OF    VOLUME    BY    TENSION.  75 

unity,  to  produce  the  same  elongation  we  should  increase  the  force 

^T 

in  the  same  ratio  as  the  two  sections,  or  make  it  —-    ;  hence  we  have 


4:T  4:1  T 

e  :  I  =  —  w  :  M*  or  M  =  —  75  .     Measure  this  modulus  for  the 

-a*  Ttea  * 

various  substances  provided,  and  compare  the  results  with  those 
given  in  the  books.  Finally,  with  an  undue  load  the  wire  will  take 
a  permanent  set,  which  increases  if  the  wire  is  stretched  for  a  con- 
siderable time.  Study  the  laws  regulating  this  property  in  the 
case  of  lead,  in  which  the  set  is  very  marked. 

34.     CHANGE  OP  VOLUME  BY  TENSION. 

Apparatus.  A  rubber  tube  AB,  Fig.  30,  about  a  metre  long, 
and  two  or  three  centimetres  in  diameter,  is  closed  above  and  be- 
low by  plugs.  The  upper  one  is  perforated,  and  carries  a  glass  tube 
with  graduated  scale  attached.  A  scale  is  also  placed  by  the  side 
of  the  rubber  tube,  and  a  number  of  points  are  marked  on  the  lat- 
ter. A  cord  and  friction  pin  C  (like  that  of  a  violin)  is  fastened  to 
the  lower  plug,  by  which  the  tension  of  the  tube  may  be  varied. 
On  the  other  side  of  the  scale  is  a  square  rod  DE  of  elastic  rub- 
ber, about  the  size  of  the  tube,  and  similarly  marked.  A  pair  of 
outside  calipers  capable  of  measuring  objects  as  large  as  the  rod  to 
within  a  tenth  of  a  millimetre,  is  also  needed. 

Experiment.  The  tube  should  be  calibrated  by  weighing  it 
when  empty,  and  when  filled  with  water  to  the  zero,  or  beginning 
of  the  glass  tube,  also  when  filled  to  some  division  w,  near  its  top. 
Call  these  three  weights  to',  to",  wrrr.  Then  w"  —  w'  —  the  weight 
of  a  cylinder  of  water  just  filling  the  tube,  or  xr%  in  which  Us 
the  known  length,  and  r  the  radius.  From  the  equation  w"  —  wr 

—  Trr2/,  we  obtain  r  =  t/  --  j  -  .    The  volume  per  unit  of  length 

w"  —  wf  .    wm  —  w" 

is  --  7  -  ,  and  in  the  same  way  for  the  glass  tube  it  is  - 

Call  the  ratio  of  these  two,  or  —77  -  r'  —  =  b.      So  much  of  the 

w    —  w     n 

work  may  be  done  once  for  all.  Any  change  in  volume  of  the 
interior  of  the  tube  can  be  accurately  measured  by  noting  the 
change  of  level  in  the  glass  tube. 

Fill  the  tube  with  water  to  the  point  marked  n,  and  read  the 
position  of  the  marks.  Stretch  it  by  turning  the  pin  below  so  as 


76 


CHANGE    OF    VOLUME    BY    TENSION. 


to  lengthen  the  tube.  Read  each  mark  in  its  new  position  to- 
gether with  the  water  level,  and  so  proceed,  taking  a  number  of 
readings  under  different  tensions. 

Construct  a  curve  in  which  abscissas  represent 
readings  of  the  water  level,  and  ordinates  the 
changes  which  take  place  in    the  length  of  the 
rubber,  draw  also  other  curves,  in  which  abscissas 
represent   the   water   level,   and    ordinates    the 
change  of  lengths  of  each  section  of  the  tube. 
To  do  this  it  is  most  convenient  to  make  a  table 
giving    the   readings   of  each    point,   a    second 
3     giving  the  difference  of  reading  of  each  two  con- 
Fig,  so.  secutive  marks,  and  a  third  giving  the  increase 
of  length  they  undergo  when   the   cord  is  stretched.     The   first 
of  the    curves  will   be  nearly  a   straight  line,  and   the  tangent 
of  the  angle  it  makes  with  the  horizontal  line,  or  the  ratio  of  its 
vertical  to  its  horizontal  progression  multiplied  by  #,  gives  the  ratio 
of  the  increase  of  volume  compared  with  the  increase  of  length. 
In  the  same  way  by  the  other  curves,  determine  the   change  in 
length  of  each  section  of  the  tube  compared  with  the  whole  change. 
Next  measure  the  height  of  each  marked  point  of  the  rubber 
rod,  also  its  diameter  at   these   points.     Stretch  it    and  measure 
again,  and  take  four  series  of  observations  in  this  way.     Now  con- 
struct curves,  in  which  abscissas  represent  scale  readings,  and  or- 
dinates alterations  in  thickness  as  given  by  the  gauge.     The  scale 
for   the  latter  must  be  greatly  enlarged.     Measure   the  area  en- 
closed by  this  space,  and  reduce  it  to  square  millimetres,  allowing 
for  the  change  of  scale.     Multiplying  this  area  by  four  times  the 
thickness  gives  approximately  the  diminution  in  volume  due  to  the 
contraction   in  the  centre.     If  the  rod  is  much  altered   in  form, 
the  change  in  cross-section  may  be  obtained  more  accurately  by 
taking  the  difference  of  the  squares  of  the  thickness  before  and 
after  extension.      Using  them  as  ordinates  of  the  curve  the  vol- 
ume is  given  by  the  enclosed   area.     Construct  such  a  curve  for 
each  extended  position  of  the  rod,  and  compare  the  decrease  of 
volume  thus  found  with  the  increase  due  to  the  change  of  length, 
or  the  product  of  the  cross-section  by  the  change  of  reading  of 
the  lower  index  mark. 


DEFLECTION    OF    BEAMS.  77 

35.     DEFLECTION  OF  BEAMS.     I. 

Apparatus.  In  Fig.  31,  AB  is  a  rectangular  bar  of  steel  rest- 
ing on  two  knife-edges,  with  a  load  applied  to  its  centre  by  a 
weight  placed  in  the  scale-pan  D.  To  measure  the  flexure,  a  mi- 
crometer screw  C  is  placed  over  the  bar,  and  turned  until  its  point 
touches  the  latter.  JZisa  galvanic  battery  having  one  pole  con- 
nected with  the  bar,  the  other  with  C  through  an  electro-magnet 
F..  When  the  screw  touches  the  bar  the  circuit  is  completed,  and 
the  magnet  draws  down  its  armature  with  a  click.  This  gives  a 
very  accurate  test  of  the  exact  position  of  the  screw  when  contact 
takes  place.  The  length  of  the  beam  may  be  altered  by  changing 
the  position  of  A  and  J5.  C  and  D  are  also  movable,  and  a  set 
of  weights  is  provided  to  vary  the  deflection.  To  ensure  contact 
the  wire  should  be  soldered  in  position,  and  the  point  of  C  tipped 
with  a  piece  of  sheet  platinum.  A  convenient  size  of  bar  for  lab- 
oratory purposes  is  about  half  a  metre  long,  a  centimetre  wide, 
and  half  a  centimetre  thick,  using  weights  from  100  to  2000 
grammes.  Instead  of  the  electro-magnet  F,  a  galvanometer  may 
be  used  if  preferred. 

Experiment.  Set  A  and  IB  50  cm.  apart,  and  C  and  D  midway 
between  them.  Turn  C  until  it  touches  the  bar,  when  instantly  a 
current  will  pass  from  the  battery 
E  through  the  magnet  F,  making 
a  click,  or  if  a  galvanometer  is  used, 
swinging  the  needle  to  the  right  or 
left.  Read  the  position  of  the  mi- 
crometer screw,  taking  the  whole 
number  of  turns  from  the  index  on 
one  side,  and  the  fraction  from  the 
graduated  circle.  Add  1000  gram- 
mes to  jf>,  and  bring  the  screw  again 
in  contact.  The  difference  in  the 

readings  gives  the  deflection  with  great  accuracy.     The  general 
formula  for  the  deflection  a  of  a  beam  of  length  I,  breadth  #,  and 

TP78 
depth  <#,  under  a  load  TFJ  is  a  —  TCTS  >  in  which  D  is  the  modu- 


lus of  transverse  elasticity,  or  the  weight  required  to  make  a 
equal  one,  when  b,  d  and  £,  all  equal  unity.  From  this  formula 
the  following  laws  may  be  deduced.  1st.  The  deflection,  when 
small,  is  proportional  to  the  weight  applied.  Measure  the  deflec- 


78 


DEFLECTION    OF    BEAMS. 


tion  for  every  hundred  grammes,  from  zero  to  two  kilogrammes, 
and  see  if  the  beam  returns  to  its  original  position  when  the  load 
is  removed.  If  not,  the  change  is  the  permanent  set.  Construct 
a  curve  with  abscissas  proportional  to  the  load,  and  ordinates  to 
the  corresponding  deflection.  Evidently,  according  to  the  law, 
this  should  be  a  straight  line,  and  the  near  agreement  proves  con- 
clusively its  correctness.  2d.  The  deflection  is  proportional  to 
the  cube  of  the  length.  Measure  the  deflection  with  a  load  of 
2  kgs.,  changing  the  length  of  beam  5  cm.  at  a  time,  from  50  cm. 
to  0,  keeping  O  and  D  always  at  the  middle  of  the  beam.  Care 
must  be  taken  in  each  case  to  first  measure  the  micrometer-read- 
ing when  no  load  is  applied,  as  this  point  will  vary,  owing  to  irreg- 
ularities in  the  bar  or  stand.  To  compare  the  results  with  theory, 
construct  a  curve  in  which  abscissas  represent  lengths  of  the  beam, 
and  ordinates  deflections.  According  to  the  law  this  should  be  a 
cubic  parabola,  having  the  equation  y  =  axs.  To  find  the  value 
of  a,  suppose  one  of  the  earlier  readings  gave  a  deflection  of 
5.8  mm.,  for  a  length,  of  40  cm. ;  then  5.8  =  a  403,  or  a  =  .00009. 
Substituting  this  value  in  the  equation,  construct  the  curve  y  = 
.00009  x*  on  the  same  sheet  with  the  experimental  curve,  and  see  if 
they  agree.  To  find  the  value  of  J),  draw  a  line  nearly  coincident 
with  the  first  series  of  observations.  Deduce  from  it  the  increase 
of  deflection  for  each  added  kilogramme.  Substitute  this  value  for 
a  in  the  formula,  making  TF—  1,  and  giving  £,  b  and  d,  their  proper 
values,  found  by  measuring  the  bar.  D  is  now  the  only  unknown 
quantity,  and  may  be  obtained  by  solving  the  equation. 

If  desired,  bars  of  different  materials  may  be  provided,  and  the 
modulus  of  each  determined  by  measuring  the  deflection  with  a 
given  load,  and  substituting  these  values  in  the  formula.  The  law 
that  the  deflection  is  inversely  proportional  to  the  breadth,  may  be 
proved  with  bars  alike  except  in  breadth,  and  the  law  of  the  thick- 
ness in  a  similar  manner.  The  form  of  the  beam  when  bent  is  found 
by  shifting  the  micrometer  G  and  measuring  the  deflection  at  vari- 
ous points  between  A  and  JB,  and  the  effect  of  a  change  in  position 
of  the  load  by  moving  D.  The  same  apparatus  may  be  applied  to 
the  case  of  a  beam  supported  at  more  than  two  points,  or  to  beams 
built  in  at  one  or  both  ends.  This  experiment  may  also  be  almost 
indefinitely  extended  by  using  circular  and  triangular  bars,  hollow 


DEFLECTION    OF    BEAMS.*  79 

or  solid,  also  those  of  a  T  or  I  shape,  or,  in  fact,  girders  of  any 
form. 

36.    DEFLECTION  OF  BEAMS.    II. 

Apparatus.  AB,  Fig.  32,  is  the  bar  to  be  tested,  which  may  be 
one  of  the  square  rods  described  in  the  next  experiment.  It  is 
clamped  at  one  end  by  placing  it  between  two  similar  rods,  one  of 
which  is  nailed  to  the  wall,  and  the  other  pressed  down  upon  it  by 
two  or  three  clamps,  like  those  of  a  quilting  frame,  as  at  O.  A 
small  mirror  maybe  placed  upon  it  at  any  point,  and  the  deflection 
measured  by  reading  in  it  the  reflection  of  a  scale  F  by  the  tel- 
escope E.  The  beam  is  bent  by  weights  placed  in  the  scale-pan  D. 
By  attaching  a  finely  divided  scale  to  A,  the  absolute  deflection 
may  be  read  by  the  telescope  E,  using  it  like  a  cathetometer. 

Experiment.  Clamp  AB  so  that  its  length  shall  be  50  cm.,  and 
place  the  mirror  at  its  end.  Place  the  telescope  E  opposite  A  and 
focus  it,  so  that  on  looking 
through  it  the  image  of 
the  scale  F  shall  be  dis- 
tinctly seen  reflected  in 
the  mirror.  Read  the  po- 
sition of  the  cross-hair  in 
the  telescope  to  tenths  of 
a  division  of  the  scale. 
Now  place  a  kilogramme 

in  D.  The  beam  is  at  once  bent,  and  although  the  mirror  moves 
but  little,  the  scale-reading  is  greatly  altered.  The  work  described 
under  the  last  experiment  may  be  repeated  with  this  apparatus. 
Or  to  vary  it,  let  the  following  measurements  be  made.  Place  2 
kgs.  in  J>,  and  take  the  scale-readings  when  placed  at  distances  of 
5,  10,  15,  20,  &c.,  centimetres  from  B.  Again,  take  the  scale-read- 
ing very  carefully  when  no  load  is  applied.  Now  place  in  D  as 
great  a  weight  as  the  beam  will  safely  bear,  and  take  readings 
every  half  minute  for  five  or  ten  minutes.  Then  remove  the  load 
and  see  if  the  beam  returns  to  its  original  position.  The  perma- 
nent set  of  the  beam  may  be  studied  in  this  manner. 

To  compare  the  results  with  theory  the  scale-readings  must  be 
reduced  to  angular  deviations  by  the  formula  given  on  page  24. 
For  small  deflections,  however,  it  is  sufficient  to  divide  the  change 


80  TRUSSES. 

in  scale-reading  by  twice  the  distance  EA^  to  obtain  the  tangent 
of  the  angle  through  which  the  mirror  moves.  In  deducing 
the  form  of  an  elastic  beam  by  Analytical  Mechanics,  we  have 


El  -7-3   =  P(l  —  #),  the  moment  of  the  deflecting  force  P.     In 

this  E  is  the  modulus  of  elasticity,  I  the  moment  of  inertia  of  the 
cross-section,  y  the  deflection  of  any  point  at  a  distance  x  from 
the  end,  and  I  the  length  of  the  beam.  Integrating,  we  obtain 
dil  JP  x2  dii 

dx  =   ~EI^X  —  "2")'  *n  wn*cn  d£"  mav  ke  compared  directly  with 

the  measurement. 

37.    TRUSSES. 

Apparatus.  A  number  of  deal  rods,  as  nearly  alike  as  possible, 
half  an  inch  square  and  five  or  six  feet  long.  These  are  to  form 
the  beams  or  units  of  which  all  the  trusses  are  to  be  composed. 
They  may  be  connected  by  clamps  like  those  used  for  quilting 
frames,  by  boring  holes  in  them  and  fastening  by  wire,  or  better 
still  by  using  small  carriage:bolts,  \"  in  diameter.  A  scale-pan  and 
set  of  weights  serve  to  apply  a  strain  not  exceeding  one  or  two 
hundred  pounds  to  any  part  of  the  truss  to  be  tested.  By  attach- 
ing a  fine  scale  the  deflection  of  any  point  may  be  read  by  a  tel- 
escope mounted  like  a  cathetometer.  The  strain  on  any  portion 
may  be  determined  by  inserting  a  small  spring  balance,  as  will 
be  described  below. 

Experiment.  This  apparatus  may  be  procured  at  very  small  ex- 
pense, while  with  it  almost  all  the  laws  of  elasticity  may  be 
proved,  and  the  strength  of  a  great  variety  of  trusses  for  bridges 
and  roofs  tested.  Although  the  following  work  resembles  that  of 
Experiment  35,  yet  its  importance,  and  the  different  method  of 
measurement  employed,  justifies  its  repetition. 

The  flexure  of  a  beam  is  proportional  to  the  load.  Set  two  knife- 
edges  40  inches  apart,  cut  a  rod  a  little  longer  than  this,  and  lay  it 
on  them.  Attach  a  fine  scale  to  its  centre  point,  focus  the  tel- 
escope on  it  and  record  the  reading.  Add  weights,  a  pound  at  a 
time,  until  the  beam  breaks.  The  increase  of  reading  in  each  case 
over  that  given  at  first,  is  proportional  to  the  load.  It  must  be 
noticed,  however,  that  when  the  beam  is  much  bent  a  new  law 
holds.  Repeat  these  measurements  with  rods  30,  20,  and  1  0  inches 


TRUSSES.  81 

long.     The  flexure  is  proportional  to  the  cube,  the  breaking  weight 
inversely  as  the  square  of  the  length. 

In  the  same  way  the  effect  of  applying  a  load  at  different  points, 
or  the  deflection  at  different  points  with  a  given  load,  may  be 
measured.  A  long  beam  may  also  be  supported  at  several  points, 
and  the  effect  of  a  moving  load  noted.  To  measure  the  strength 
of  a  beam  built  in  at  one  or  both  ends,  clamp  it  at  those  points 
between  two  similar  beams,  one  above,  the  other  below,  and  fasten 
them  by  bolts  or  clamps  to  a  fixed  upright.  Having  thus  fully 
determined  the  strength  of  the  single  beams,  they  should  be  ex- 
amined when  combined.  Join  two  beams  together  so  as  to  form  a 
T,  and  measure  the  strain  necessary  to  pull  the  vertical  one  off. 
Different  kinds  of  joints  may  thus  be  compared.  Some  form  of  truss 
should  now  be  built,  and  its  strength 
tested.  Let  the  king-post,  Fig.  33,  be 
the  form  selected,  and  make  the 
span  AS  40  inches.  Cut  two  rods 
a  little  longer  than  this,  and  bore  the 

holes  A,  B  and  D.     Cut  two  more  , 

rods    CD  with,  holes    distant    10 

inches,    and    attach    them    to    the  rig  ^ 

others  by  bolts   at   D.    Add  AO 

and  CB,  and  connect  the  two  trusses  thus  formed  by  cross  pieces 

at  A,  B,  C  and  Z>,  so  that  they  shall  be  ten  inches  apart.     A 

small  bridge   is  thus  made  whose  stiffness   is   remarkably  great. 

Such  a  structure  should  bear  a  weight  of  fifty,  or  even  a  hundred 

pounds,  with  little  flexure.     Measure  the  deflection  at  different 

points  under  varying  loads,  trying  the  effect  of  applying  the  latter 

at  the  centre,  on  one  side,  or  distributing  it  uniformly. 

The  force  acting  on  any  beam  is  readily  determined  by  replacing 
it  by  a  spring  balance,  with  a  screw  and  nut  which  may  be  made 
to  take  up  the  whole  strain,  without  distorting  the  structure. 
If  the  force  is  one  producing  compression,  it  is  best  to  elongate  the 
beams,  as  at  (ZEJ  and  insert  the  balance  between  C  and  JS.  Com- 
pare the  various  results  with  those  obtained  by  computation.  It 
must  be  remembered  that  this  method  of  connecting  the  beams  of 
a  truss  by  bolts  is  not  employed  in  actual  practice,  but  it  is  very 
convenient,  and  sufficiently  strong  for  a  model.  If  preferred, 
6 


82  LAWS    OF    TORSIOX. 

proper  tools  may  be  supplied,  and  the  student  may  frame  his  truss 
as  in  a  real  bridge  or  roof.  Further,  any  beam,  as  CD,  subjected 
only  to  tension,  may  be  replaced  by  a  wire. 

The  rods  will  also  be  found  useful  for  a  great  variety  of  pur- 
poses. Thus  one  of  the  easiest  ways  to  make  large  screens  is  to 
fasten  four  of  them  together  at  the  ends,  and  attach  thick  paper  to 
them  by  double-pointed -tacks.  Again,  a  convenient  way  to  make 
a  large  rectangular  box  to  cut  off  light,  or  to  protect  an  instru- 
ment from  dust,  is  to  connect  twelve  of  these  rods  at  the  ends  by 
slipping  them  into  corner  pieces  of  tin,  Fig. 
34,  and  covering  them  with  black  paper. 
In  the  same  way  all  the  principles  of 
framing  may  be  taught,  and  quite  compli- 
cated structures  built.  It  is  well  to  have 
some  of  the  latter  loaded  until  they  break, 
to  determine  the  weakest  points.  They  are 
then  easily  repaired  by  inserting  new  pieces 
of  wood.  Another  very  good  object  to  construct  and  test  is  a  sus- 
pension bridge.  Use  two  stout  wires  or  chains  for  the' suspension 
cables,  and  build  the  roadway  of  the  rods,  hanging  it  from  the 
chains  by  wires  with  screw  threads  cut  on  their  ends,  so  that  their 
lengths  may  be  adjusted  by  nuts.  Test  the  strain  on  the  chains 
by  inserting  a  spring  balance  like  that  known  as  the  German  ice- 
balance,  and  measure  the  deflections  of  the  different  parts  under 
varying  loads. 

38.    LAWS  OF  TORSION. 

Apparatus.  Let  AB,  Fig.  35,  represent  the  bar  whose  torsion 
is  to  be  measured.  The  farther  end  B  is  firmly  fastened  to  a 
piece  of  wood  (7,  which  can  turn  around  the  axis  of  the  beam,  but 
may  be  clamped  in  any  position  to  the  semicircle  immediately  be- 
hind it.  A  in  the  same  way  is  attached  to  J9,  which  carries  two 
brass  rods,  acting  like  the  dog  of  a  lathe.  EF  is  a  long  rod 
mounted  on  an  axle,  which  may  be  turned  by  placing  weights  in 
the  scale-pan  Q.  The  latter  is  supported  by  a  cord  passing  over  a 
curved  block  at  the  end  of  E,  so  that  the  moment  of  the  weight, 
or  its  tendency  to  twist  the  bar  is  unchanged,  whatever  the  posi- 
tion of  EF.  To  measure  the  angle  of  torsion,  two  mirrors  are 
attached  to  AJB,  and  the  scales  H,Hf  reflected  in  them  are  viewed 
by  telescopes  T,  I'.  By  making  H,  H!  arcs  of  circles,  with  centres 
at  A  and  B,  the  angle  of  torsion  may  be  obtained  directly. 


LAWS    OF    TORSION. 


83 


Fig.  35. 


Experiment.  To  measure  the  torsion  of  the  beam  AB,  attach 
it  to  the  stand,  as  in  the  figure,  placing  the  mirrors  at  a  distance 
apart  equal  to  the  length 
to  be  examined.  Focus 
the  telescopes  I,  I'  so 
that  the  scales  H,  Hf 
shall  be  distinctly  visible, 
and  read  the  position  of 
the  cross -hairs.  Place  a 
weight  in  the  scale-pan 
6r  which  will  twist  both 
the  mirrors  A  and  J?,  de- 
viating the  former  the 
most.  See  how  much 
each  scale-reading  has 
changed ;  their  difference 
measures  the  angular 
twist  of  AS.  This  may  be  reduced  to  degrees  from  the  radius 
of  curvature  of  the  scale,  and  the  magnitude  of  its  divisions,  recol- 
lecting that  the  motion  of  the  mirror  is  only  one-half  that  of  the 
reflected  ray.  From  the  theory  of  torsion  the  following  laws  are 
readily  deduced.  The  angle  of  torsion  is  proportional  to  the  mo- 
ment of  the  deflecting  force.  To  prove  this  law,  measure  the  tor- 
sion with  several  different  weights  in  6r,  and  see  if  the  angles  are 
proportional  to  the  weight.  The  distance  of  the  point  of  applica- 
tion of  Gr  from  the  axis  may  also  be  varied,  and  it  will  be  found 
that  the  torsion  is  proportional  to  the  product  of  this  distance  mul- 
tiplied by  6r.  The  torsion  is  proportional  to  the  length  of  the  bar. 
Prove  this  by  varying  the  distance  between  the  mirrors,  leaving 
the  bar  unchanged.  By  using  a  variety  of  bars  it  may  be  proved 
that  in  those  having  similar  sections  the  torsion  is  proportional  to 
the  fourth  power  of  the  similar  dimensions.  In  rectangular  bars  of 
breadth  £,  and  depth  c?,  it  is  proportional  to  -^bd  (b2  +  (#2),  and  in 
tubes  to  ^7r(>4  —  /4)  calling  r  and  /  the  outer  and  inner  radii.  In 
practice,  owing  to  the  warping  of  the  surfaces,  these  formulae  un- 
dergo slight  modifications. 


04:  FALLING  BODIES. 

39.    FALLING  BODIES. 

Apparatus.  This  consists  of  two  parts,  a  chronograph  capable 
of  measuring  very  minute  intervals  of  time,  as  hundredths  of  a 
second,  and  the  arrangement  represented  in  Fig.  36,  for  making 
and  breaking  an  electric  circuit  when  the  body  falls.  A  ball  A  is 
attached  to  the  spring  D  by  a  short  thread  and  wire.  Burning 
the  thread  the  ball  is  released,  and  the  spring  rising  allows  the  cur- 
rent to  pass  from  the  battery  B,  to  (7,  Z>,  E,  F,  and  the  chronograph 
G.  This  marks  the  beginning  of  the  time  to  be  recorded.  Its 
end  is  shown  by  the  breaking  of  the  circuit,  which  occurs  at  F 
when  A  strikes  E.  To  have  the  current  broken  during  this  time, 
instead  of  closed,  it  is  merely  necessary  to  place  the  points  E  and 
F  below  the  springs.  Another  method  of  releasing  the  ball  is  to 
put  an  iron  pin  in  its  upper  side,  and  support  it  by  an  electro- 
magnet. The  instant  the  current  is  broken  it  will  fall.  One  of 
the  best  forms  of  chronograph  for  this  purpose  is  that  devised  by 
Hipp,  in  which  a  reed  making  a  thousand  vibrations  a  second  re- 
places the  pendulum  of  an  ordinary  clock.  It  therefore  ticks  a 
thousand  times  a  second,  and  measures  small  intervals  of  time  with 
great  precision.  Very  good  results  may  be  obtained  with  a  com- 
mon marine  clock,  removing  the  hairspring  and  replacing  it  by  an 
elastic  bar  of  steel.  An  electro-magnet  is  placed  close  to  the  pallet, 
so  that  when  the  current  passes  the  spring  is  bent.  The  clock, 
therefore,  starts  the  instant  the  circuit  is  broken,  and  stops  as  soon 
as  it  is  closed. 


Experiment.    The  time  of  falling  of  any  body  through  a  height 
equals  y— ,  in  which  g  =  9.80  metres.    Measure  the  time  of 

Y  y 

fall  through  various  heights  by  altering 
the  length  of  the  wire  AD,  noting  the 
height  in  each  case  with  care,  repeating 
several  times  and  taking  the  mean.  Com- 
pare this  with  the  result  given  by  theory. 
In  vacuo  the  time  would  be  independent 
of  the  material  or  magnitude  of  the  ball. 
In  air,  however,  this  is  not  the  case,  owing 
to  the  resistance.  The  latter  may  there- 
fore  ke  determined  by  measuring  the  time 
of  fall  of  bodies  having  the  same  form  but  different  weights.  This 
apparatus  may  also  be  applied  to  the  measurement  of  the  velocity 
of  curve-motion  and  personal  equation,  or  the  coefficient  of  friction 


METRONOME    PENDULUM.  85 

of  a  body  may  be  found  with  great  accuracy  by  measuring  its  time 
of  descent  down  an  inclined  plane. 

40.    METRONOME  PENDULUM. 

Apparatus.  A  light  deal  rod  is  provided,  to  which  two 
leaden  weights  may  be  attached  at  any  point  by  set  screws.  A 
knife-edge  passes  through  the  centre  of  the  rod,  so  that  it  may  be 
swung  like  a  pendulum.  Either  weight  may  also  be  attached  to 
the  knife-edge  by  a  fine  wire.  A  millimetre  scale  is  used  to  meas- 
ure the  distance  of  the  weights  from  the  knife-edge,  and  a  bracket 
fastened  to  the  wall  and  carrying  a  steel  plate  serves  as  a  support. 

Experiments.  First,  to  prove  the  law  of  the  single  pendu- 
lum. Attach  the  heavier  weight  to  the  knife-edge  by  the  wire, 
using  as  great  a  length  as  possible.  Measure  the  time  in  sec- 
onds of  making  100  single,  or  50  double  vibrations,  also  the  dis- 
tance from  the  centre  of  the  lead  to  the  knife-edge.  Repeat  with 
several  lengths  of  wire. 

Now  compare  the  observed  time  with  that  given  by  the  formula 

jl 
t  =  TT  t/-,  in  which  g  =  9.80  m.,  and  I  is  the  measured  length. 

T     t/ 

Repeat  one  of  these  measurements  with  the  lighter  weight,  using 
the  same  value  of  I.  It  should  give  the  same  result  as  the  other. 
Next  place  both  weights  on  the  deal  rod  at  opposite  ends.  Meas- 
ure, as  before,  their  time  of  vibration,  also  their  distances  from  the 
knife-edge.  Compute  the  time  of  vibration  as  above,  merely  sub- 

20  7' 2   I  'w"lffz 
stituting  for  I  the  value  — /,/   , — rnrri  in  which  «/,  w"  are  the 

W  i     ~\~   W     I/ 

weights,  and  Z',  I"  their  distances  from  the  knife-edge.  If  greater 
accuracy  is  required,  a  third  term  must  be  introduced  for  the 
weight  of  the  rod.  Repeat  with  various  positions  of  the  two 
weights,  and  compare  the  results  with  theory. 

41.     BORDA'S  PENDULUM. 

Apparatus.  In  Fig.  37,  CD  is  the  pendulum,  formed  by  attach- 
ing a  ball  of  lead  C,  to  a  wire  nearly  four  feet  long,  and  supporting 
it  on  a  knife-edge  D.  A  sheet  of  platinum  is  fastened  below  the 
ball,  so  that  when  at  rest  it  dips  edgewise  into  a  mercury  cup, 
making  electrical  connection  with  the  battery  B.  E  is  a  clock 
whose  pendulum  F  dips  in  a  second  mercury  cup.  When  both 
pendulums  are  at  rest  the  current  passes  from  B  through  (7,  J),  E 


86 


BORDA S    PENDULUM. 


and  2>\  to    6r,  which  is  an  electric  bell  arranged  so  as  to  strike 
whenever  the  circuit  is  closed. 


Experiment.     Connect  the  battery  IB  with  the  wires  attached  to 
C  and    G-.     The  bell  will  instantly  strike.     Start  the   pendulum 
D  C.     Whenever  it  passes  through  the  mercury  cup,  that  is,  with 
every  swing,  the  electric  current  passes  through  G  and  makes  the 
bell  strike.     Stop  CD  and  start  the  clock.     The  strokes  now  occur 
at  intervals  of  exactly  one  second.     Now  set  both  pendulums  go- 
ing.    The  bell  will  strike  only  when 
both  are  vertical  at  the  same  instant. 
This  will  occur  at  regular  intervals, 
equal  to  the  time  required  by  the 
longer  pendulum  to  lose  just  one 

fijyjuN  vibration.     Record  the  minute  and 

(JUT  \  second  of  each  stroke  for  ten  or  fif- 

teen minutes  consecutively.  The 
first  differences  give  the  intervals, 
and  from  the  mean  of  the  latter  the 
time  of  vibration  may  be  computed 
with  great  accuracy.  For  exam- 
ple, if  the  interval  is  47  seconds  it 

denotes  that  in  this  time  CD  made  one  less,  or  46  vibrations,  hence 
the  time  of  a  single  vibration  would  be  f$=1.0217  seconds.  An 
error  of  one  second  in  the  mean  of  the  interval  would  make  the 
time  £f=1.0213  seconds,  or  alter  the  result  less  than  a  two-thou- 
sandth of  a  second.  The  method,  therefore,  is  one  of  extreme  pre- 
cision. Sometimes,  especially  when  the  pendulum  is  swinging 
through  4  small  arc,  the  bell  will  strike  for  several  consecutive  sec- 
onds, owing  to  the  considerable  interval  of  time  during  which  con- 
tact is  made  at  C",  so  that  for  several  seconds  the  circuit  is  closed 
at  F  before  it  is  broken  at  C.  In  this  case  the  time  of  the  first 
stroke  should  be  recorded  and  their  number ;  the  true  time  being 
taken  as  the  mean  of  the  first  and  last.  To  make  CD  vibrate  in 
one  plane  instead  of  describing  an  ellipse,  attach  a  fine  thread  to 
the  ball  (7;  draw  it  to  one  side  about  ten  inches;  let  it  come  to 
rest,  and  then  burn  the  thread.  Finally  measure  the  length  I  of  the 
pendulum,  or  the  distance  from  the  knife-edge  to  the  centre  of  the 


Fig.  37. 


i/ 

T 


TORSION    PENDULUM.  87 

ball,  and  compute  the  force  of  gravity  g  from  the  formula  ;     t  = 
—,  in  which  t  equals  the  time  of  vibration,  and  TT  —  3.1416. 

y 

This  experiment  may  be  repeated  with  a  different  length  of  pen- 
dulum, or  it  may  be  varied  so  as  to  prove  that  the  time  increases 
with  the  amplitude.  In  the  latter  case  the  arc  through  which  the 
ball  swings  should  be  as  large  as  possible,  and  it  should  be  meas- 
ured as  it  progressively  diminishes.  To  compute  the  theoretical 
time  of  swinging  through  any  arc  a,  divide  versin  \  a,  or  the  ver- 
tical distance  through  which  the  ball  moves,  by  its  length,  and  call 
the  quotient  x.  Then  the  time  if  for  any  value  of  x  may  be  found 
from  the  equation  if  =  (I  +  |  x  +  ITS  &  +  &c.)  £,  in  which  t  is 
the  time  when  the  arc  is  very  small.  When  a  =  180°,  or  the  ball 
swings  through  a  semicircle,  t'  =  1.180  £,  when  a  =  30°,  if  = 
1.0063  t,  when  a  =  10°,  if  =  1.00067  t,  hence  for  small  arcs  the 
correction  for  this  cause  is  very  small.  If  great  accuracy  is  re- 
quired in  this  experiment  the  suspending  wire  should  be  very 
light,  and  with  the  knife-edge  should  vibrate  in  about  one  second 
when  the  ball  is  removed,  or  a  correction  may  be  applied  for  them 
as  described  in  Experiment  40. 

42.     TORSION  PENDULUM. 

Apparatus.  AB,  Fig.  38,  is  a  vertical  wire  with  an  index  C, 
which  moves  over  a  graduated  circle.  Weights  of  a  cylindrical 
form,  as  .Z>,  may  be  attached  below  in  such  a  manner  that  the 
wire  cannot  twist  without  turning  them.  To  vary  the  length  of  the 
wire  it  is  passed  around  several  small  brass  tubes  E,  F,  G-,  placed 
at  different  heights,  so  that  it  may  be  clamped  at  these  points  by 
inserting  a  pin  G-  passing  into  a  hole  bored  behind  them.  A  scale 
and  clock  beating  seconds  are  also  needed  for  this  experiment. 

Experiment.  1st.  The  time  is  indepen- 
dent of  the  amplitude.  of  the  vibration.  Use 
the  whole  length  of  the  pendulum,  and  apply 
such  a  weight  that  the  time  of  a  single  vi- 
bration shall  be  about  one  second.  Twist 
the  index  through  a  small  arc,  and  take  the 
time  of  one  hundred  oscillations  by  noting 
the  position  of  the  index  at  the  beginning  of 
a  minute,  and  the  exact  time,  when  after 
making  one  hundred  single  oscillations,  it 


88 


TORSION    PENDULUM. 


again  reaches  the  same  point.  Dividing  the  interval  by  one  hun- 
dred gives  the  time  of  a  single  oscillation.  Repeat  two  or  three 
times  with  arcs  of  different  magnitudes,  and  compare  the  results. 
2d.  The  time  is  proportionate  to  the  length  of  the  wire.  Make 
the  same  experiment,  first  with  the  wire  of  its  full  length,  then, 
passing  the  pin  through  the  different  tubes  E,  F,  clamping  it  at 
these  points.  Measure  their  distances  from  J5,  and  compare  with 
the  law.  In  the  same  way  the  relation  of  the  time  to  the  diameter 
of  the  weight,  or  to  its  length,  may  be  tested  and  compared  with 
theory. 


MECHANICS  OP  LIQUIDS  AND  GASES. 


43.     PRINCIPLE  OP  ARCHIMEDES. 

Apparatus.  An  inverted  receiver  J.,  Fig.  39,  with  a  stopcock, 
or  better,  an  J"  gas  valve  below.  Near  the  top  is  placed  a  hook 
C  with  a  sharp  point,  which  is  used  to  mark  the  level  of  the  liquid. 
The  whole  may  be  hung  from  the  scale-pan  D  of  a  large  balance, 
EF,  which  has  a  counterpoise  attached  to  the  other  end.  G  is  a 
beaker  to  collect  the  water  drawn  off,  and  H  a  stand  by  which  A 
may  be  supported  if  necessary.  A  set  of  weights  is  needed,  also 
two  bodies  M  and  JVJ  one  heavier,  the  other  lighter  than  water. 
They  may  be  made  of  metal  and  wood,  or,  if  preferred,  of  glass, 
and  loaded  so  that  one  shall  float,  the  other  sink. 

Experiment.  1st.  A  heavy  body  when  immersed  is  buoyed  up 
by  a  force  equal  to  the  weight  of  the  displaced  liquid.  Place  the 
receiver  on  the  stand,  fill  it  with  water  and 
draw  out  the  latter  until  the  point  of  the  hook 
just  touches  the  surface,  observing  the  point 
of  contact,  as  in  Experiment  13.  Place  the 
beaker  Gr  on  the  scale-pan  J>,  suspend  M  be- 
low it,  and  add  weights  to  the  other  side  so 
as  to  bring  the  beam  into  equilibrium.  If 
now  the  receiver  is  brought  up  under  M  the 
water  will  rise,  and  the  equilibrium  will  be 
destroyed.  Open  £  and  draw  off  the  water 
into  6r  until  M,  being  immersed,  the  level  is 
again  exactly  at  C.  Now  replacing  Gr  on  D 
it  will  be  found  that  the  equilibrium  is  re- 
stored. Hence  the  loss  of  weight  of  JJf  equals  K  ^ 
the  gain  of  6r,  or  the  weight  of  the  displaced 
liquid,  since  the  level  is  unchanged. 

2d.     Since  action  and  reaction  are  equal,  the  vessel  appears  to 


90  RELATION    OF    WEIGHTS    AND    MEASURES. 

gain  in  weight  by  an  amount  just  equal  to  the  loss  of  M.  Sus- 
pend A  from  the  scale-pan,  and  M  from  the  stand.  Bring  the 
water-level  to  C  and  counterpoise  as  before.  Immerse  M,  when 
the  water  will  rise,  and  the  weight  apparently  increase.  Open  B 
therefore,  and  draw  out  the  water  until  the  level  is  restored,  when 
it  will  be  found  that  the  beam  is  again  balanced,  showing  that  it 
was  necessary  to  draw  out  a  volume  of  liquid  equal  to  that  of  M. 

3d.  A  floating  body  displaces  a  weight  of  liquid  just  equal  to^ 
its  own.  Rest  A  on  its  stand  and  restore  the  water  level  to  C. 
Place  G  and  JV"  on  the  scale-pan  and  counterpoise.  Let  N  float  in 
A,  open  IB  until  the  proper  level  is  attained,  collect  the  water  in 
G,  and  replacing  the  latter  on  the  scale  it  will  be  found  that  the 
equilibrium  is  restored.  That  is,  the  weight  of  the  displaced  wa- 
ter, or  the  increase  of  G,  equals  the  weight  of  N". 

44.    RELATION  OF  WEIGHTS  AND  MEASURES. 

Apparatus.  A  delicate  balance  with  a  counterpoise  on  one  side 
and  scale-pan  on  the  other,  below  which  a  small  cube  of  brass  is 
suspended  by  a  very  fine  platinum  wire.  In  addition,  a  beaker 
containing  distilled  water,  a  thermometer  and  a  set  of  weights, 
must  be  provided. 

Experiment.  By  definition  a  gramme  is  the  weight  in  vacuo 
of  a  cubic  centimetre  of  distilled  water,  at  the  temperature  of 
maximum  density,  that  is,  about  4°  C. ;  the  object  of  the  present 
experiment  is  to  test  this  relation.  Add  weights  to  the  scale-pan 
until  equilibrium  is  established ;  then  immerse  the  cube  in  the  dis- 
tilled water,  first  washing  it  with  caustic  potash  to  remove  the  air, 
then  very  thoroughly  with  common  water  to  remove  the  potash, 
and  finally  with  distilled  water.  The  weight  now  required  to 
counterpoise  it  will  be  greater  than  that  previously  taken,  by  an 
amount  equal  to  the  weight  of  the  displaced  water.  Record  the 
height  of  the  barometer  and  the  temperature  of  the  water.  Next, 
to  determine  the  volume  of  the  cube,  measure  the  twelve  edges 
very  carefully  to  tenths  of  a  millimetre,  and  take  the  mean  of  each 
set  of  four  which  are  parallel.  The  product  of  these  three  means 
equals  the  volume.  The  dividing  engine  should  be  used  to  attain 
sutficient  accuracy  in  this  measurement.  Add  to  this  the  volume 
of  the  wire  found  by  multiplying  its  cross  section  by  the  length 


HYDROMETERS.  91 

submerged.  Correct  the  weight  found  above  for  the  buoyancy  of 
the  air,  and  the  volume  for  the  dilatation  of  the  water,  as  in  Ex- 
periment 19.  Only  in  this  case  the  whole  weight  of  the  displaced 
air  must  be  added,  since  by  definition  the  weight  must  be  taken  in 
vacuo.  The  density  I)  of  the  water  at  any  ordinary  temperature 
t,  is  given  by  the  formula  D  =  1  —  .000006  (t  —  4)2,  its  density 
at  4°  being  unity.  After  applying  these  two  corrections,  see  if  the 
volume  of  the  water  in  cm8  equals  its  weight  in  grammes. 

By  using  English  weights  and  measures  instead  of  French,  the 
relation  between  the  inch  and  pound  may  be  established  in  a 
similar  manner. 

45.    HYDEOMETEES. 

Apparatus.  This  consists  of  four  tall  jars,  two  containing 
water,  the  third  some  light  liquid,  as  alcohol,  and  the  fourth  a 
saturated  solution  of  salt,  or  other  heavy  liquid.  A  variety  of 
hydrometers,  some  giving  the  specific  gravity  directly,  others  with 
the  scales  of  Beaume,  Cartier  and  Beck,  &c.  In  one  of  the  jars 
of  water,  which  should  be  larger  than  the  other,  is  a  Nicholson's 
hydrometer,  Fig.  40,  and  on  the  table  a  box  of  weights,  a  small 
stone  and  a  piece  of  hard  wood.  Near  by  should  be  a  sink,  with 
a  large  jar  in  it,  through  which  water  is  continually  flowing,  to 
wash  the  hydrometers. 

Experiment.  Float  each  hydrometer  in  turn  in  the  jar  contain- 
ing water,  and  record  the  reading  of  the  point  of  the  scale  on  its 
stem  just  at  the  surface.  This  point  is  determined  most  accurately 
by  bringing  the  eye  nearly  on  a  level  with  the  top  of  the  water, 
but  a  little  below  it.  All  should  give  a  specific  gravity  of  very 
nearly  unity,  the  difference  being  partly  due  to  error  in  the  instru- 
ment, and  partly  to  expansion  of  the  water  by  heat.  Next  im- 
merse each  in  the  alcohol,  take  the  reading  and  wash  by  plunging 
it  in  the  large  vessel  of  water.  Do  the  same  with  the  solution  of 
salt.  If  any  hydrometer  sinks  lower  than  the  top  of  its  scale,  the 
liquid  is  lighter  than  it  can  measure;  if  it  floats  too  high  the  liquid 
is  too  heavy.  Finally,  reduce  all  the  readings  to  specific  gravities 
by  the  hydrometer  tables.  These  instruments  being  of  glass  are 
easily  broken,  and  must  be  handled  with  care. 

Turning  now  to  the  Nicholson's  hydrometer,  place  weights  on 
the  upper  scale-pan  A,  until  it  sinks  to  the  mark  scratched  on  its 


92 


SPECIFIC    GRAVITY    BOTTLE. 


stem.    Record  their  sum,  and  replace  them  in  their  box,  taking 
care  (as  must  always  be   done  with  delicate  weights)  never   to 
touch  them  with  the  fingers,  but  only  with  forceps.      Moreover 
they  must  never  be  laid   down  on  the  table,  and  to  prevent  their 
falling  into  the  water,  the  piece  of  metal  C  must  be  kept  over  the 
mouth  of  the  jar.     Place  the  stone,  or  other  object  whose  specific 
gravity  is  to  be  measured,  on  A,  and  add  weights,  as  before.     Call 
their  sum  in  the  first  case  w,  in  the  second  wf.     Raise 
the  hydrometer  out  of  the  water  (of  course  first  re- 
placing the  weights  in  their  box),  and  place  the  stone 
on  the  lower  scale-pan  B.    Immerse  it,  taking  care 
that  there  are  no  adhering  air  bubbles.     If  these  can- 
not be  detached  with  the  finger,  remove  the  stone  and 
wash  it  first  with  caustic  soda,  and  then  with  pure 
water.     Call  w"  the  weight  required  to  immerse  the 
hydrometer  when  the  stone  is  on  JB.     Then  w"  —  w' 
is  the  apparent  diminution  of  weight  of  the  stone  when  immersed, 
or  the  weight  of  an  equal  bulk  of  water.    As  w  —  wf  is  the  weight 

of  the  stone,  its  specific  gravity  is     //      — /.      Perform   the   same 

experiment  with  the  piece  of  wood,  only  placing  it  below  IB  to 
keep  it  down,  and  noticing  that  w"  will  be  greater  than  w. 

46.     SPECIFIC  GRAVITY  BOTTLE. 

Apparatus.  A  balance  weighing  up  to  100  grms.,  and  turning 
with  two  or  three  milligrammes,  a  set  of  weights  and  a  specific 
gravity  bottle,  or  as  a  substitute,  two  glass  stoppered  bottles,  the 
neck  of  one  being  large,  of  the  other,  small.  They  should  be  care- 
fully selected,  with  stoppers  fitting  smoothly,  and  a  scratch  should 
be  made  both  on  the  neck  and  stopper,  so  that  the  latter  may 
always  be  turned  into  the  same  position.  As  objects  for  determ- 
ination of  specific  gravity  any  liquid  may  be  used,  as  a  solution  of 
salt,  and  two  or  three  solids,  as  stones,  coins,  gold  ornaments, 
sand,  &c. 

Experiment.  Weigh  the  empty  bottle  and  stopper,  and  call 
their  weight  w&.  Fill  the  bottle  with  water,  insert  the  stopper  and 
wipe  off  the  liquid  which  has  overflowed,  taking  care  that  the  ex- 
terior of  both  bottle  and  stopper  are  perfectly  dry.  Call  this 
weight  ww. 


HYDROSTATIC    BALANCE.  93 

Fill  with  the  liquid  to  be  tested  in  the  same  way,  taking  care 
that  the  stopper  is  inserted  in  the  same  position  as  before,  and  that 
no  liquid  adheres  to  the  exterior.  Let  the  weight  be  wit  then 

w\  —  wa  and  ww  —  w&  are  the  weights  of  equal  bulks  of  the  liquid 

w\  —  ioa 
and  of  water,  and  the  specific  gravity  of  the  liquid  is  =  w   w  • 

To  find  the  specific  gravity  of  a  solid,  use  the  bottle  with  the 
larger  neck.  Call  w  the  weight  of  the  solid,  ww  the  weight  of  the 
bottle  filled  with  water,  and  we  the  weight  when  the  solid  is  in- 
serted, and  the  remaining  space  filled  with  liquid ;  then  w  +  w^  — 
w6  equals  the  weight  of  a  volume  of  water  equal  to  that  of  the 

solid,  and  the  specific  gravity  =  — -r— -  — .      This    method   is 

J       w  -f-  wv  —  io8 

applicable  to  solids  heavier  or  lighter  than  water.  The  principal 
precaution  is  to  take  care  that  no  bubbles  adhere  to  the  solid  or 
sides  of  the  bottle,  and  that  the  stopper  is  always  pressed  in  by  the 
same  amount.  Use  the  same  devices  for  removing  the  air  as  with 
the  Nicholson's  hydrometer,  Experiment  45.  With  metals  these 
precautions  are  especially  important,  or  large  errors  will  be  intro- 
duced. Another  good  method  is  to  place  the  flask  containing  the 
solid  and  water  under  the  receiver  of  an  air-pump  and  exhaust 
two  or  three  times.  This  method  is  not  applicable  to  wood,  as  it 
removes  the  air  from  the  cells,  and  increases  the  apparent  specific 
gravity.  The  same  effect  is  produced  by  long  immersion,  and 
finally  when  waterlogged,  the  specific  gravity  becomes  greater  than 
unity,  and  the  wood  sinks. 

47.    HYDROSTATIC   BALANCE. 

Apparatus.  A  complete  apparatus  for  this  purpose,  known  as 
Mohr's  Balance,  may  be  obtained,  and  the  following  description  is 
especially  applicable  to  it.  A  common  balance  may,  however,  be 
substituted,  raising  one  scale-pan  and  attaching  a  hook  below.  In- 
stead of  riders  it  is  then  generally  more  convenient  to  use  ordinary 
weights.  Some  solids  and  liquids  are  also  needed  as  substances 
whose  specific  gravity  is  to  be  determined. 

Experiment.  Attach  the  small  scale-pan  to  the  left,  and  the 
glass  counterpoise  to  the  right  end  of  the  beam.  The  weighing 
is  done  by  riders,  of  which  there  are  three  sizes,  whose  weights  are 
in  the  ratio  10,  100  and  1000.  The  beam  is  divided  into  10  equal 
parts,  so  that  when  balanced  the  weight  may  be  read  off  directly 


94  EFFLUX    OF    LIQUIDS. 

to  three  places  of  decimals.  Fill  the  small  jar  with  water,  and  see 
what  weight  is  necessary  to  immerse  the  counterpoise.  It  will  be 
found  to  be  very  nearly  1000,  and  evidently  equals  the  weight  of 
the  water  displaced.  Next,  fill  the  jar  with  the  liquid  to  be  tested, 
and  see  what  weights  are  now  required.  The  ratio  in  the  two 
cases  is  the  specific  gravity.  The  temperature  should  be  recorded 
in  each  case  by  the  thermometer  contained  in  the  counterpoise, 
and  if  great  accuracy  is  required  a  correction  applied  for  it,  or  bet- 
ter, the  liquids  may  be  cooled  to.  the  standard  temperature. 

To  find  the  specific  gravity  of  a  solid,  wind  a  piece  of  fine 
wire  around  it,  and  suspend  from  the  left  hand  end  of  the  beam. 
Counterpoise  by  adding  lead,  sand  or  paper  to  the  scale-pan  at 
the  other  end  until  a  perfect  balance  is  obtained.  Immerse  in  a 
vessel  of  water,  and  balance  by  adding  the  riders ;  their  weight 
equals  that  of  an  equal  volume  of  water.  Then  remove  the  solid, 
and  again  bring  the  beam  to  a  horizontal  position  by  the  riders  ; 
this  gives  the  weight  of  the  solid,  which  divided  by  the  weight  of 
the  water  displaced,  gives  the  specific  gravity.  If  more  convenient, 
the  weight  of  the  body  may  be  obtained  directly  by  the  riders  with- 
out counterpoising  it. 

Next,  find  the  specific  gravity  of  a  piece  of  wood,  or  other  solid 
lighter  than  water.  Attach  a  piece  of  lead,  or  other  body  heavy 
enough  to  sink  it,  and  measure,  as  above,  the  following  quantities. 
"Weight  of  solid  in  air  t0s,  weight  of  lead  in  air  t01?  weight  of  lead 
in  water  w{,  weight  of  solid  and  lead  in  water  wlsf.  Then  w\  —  w\ 
=  weight  of  a  bulk  of  water  equal  to  that  of  the  lead.  wl  +  wa 
—  wia'  —  weight  of  a  bulk  of  water  equal  to  lead  and  solid.  Hence 
their  difference,  or  wl  -f-  w6  —  wla'  —  wl  -f-  w'  =  w&  +  w{  —  wls'  = 
weight  of  water  equal  in  bulk  to  solid,  and  weight  of  solid  divided 

by  this,  equals  specific  gravity,  or  $.  G-.  =  — ^ — r r 

The  same  precautions  are  necessary,  as  with  the  gauge  flask, 
regarding  air  bubbles,  and  the  riders  should  never  be  touched  with 
the  fingers,  but  always  with  a  small  bent  wire. 

48.    EFFLUX  OF  LIQUIDS. 

Apparatus.  In  Fig.  41,  A  and  IB  are  two  reservoirs  of  tin,  or 
wooden  boxes  lined  with  lead,  each  containing  two  or  three  cubic 


EFFLUX    OF    LIQUIDS. 


95 


feet.  "Water  is  admitted  by  a  valve  at  (7,  and  passes  through  a 
cylinder  of  perforated  tin  .Z>,  to  break  up  the  stream  and  prevent 
much  motion  of  the  water  in  A.  An  outlet  is  made  at  E^  which 
may  be  closed  by  a  stick  of  wood  with  a  rubber  flap  on  its  end  J£, 
which  is  held  in  place  by  the  pressure  of  the  water.  To  keep  the 
level  constant,  a  funnel  F  is  connected  with  the  interior  by  a  rub- 
ber tube,  so  that  it  may  be  raised  or  lowered,  and  serve  as  an  over- 
flow, or  a  simple  straight  tube  may  be  used,  passing  through  the 
bottom  of  A  to  the  surface.  The  height  of  the  water  is  read  by  a 
hook  gauge  Gf  with  an  index  attached,  moving  over  a  scale.  A 
number  of  brass  plates  fitting  into  .2?  are  provided  with  orifices  of 
various  shapes  and  sizes,  some  circular,  rectangular  and  triangular, 
and  others  furnished  with  projecting  cylindrical  or  conical  tubes. 

The  second  reservoir  JS  has  also  a  hook  gauge  and  scale  H  to 
show  the  amount  of  water  in  it,  and  an  outlet  I  closed  by  a  plug. 
To  prevent  motion  of  the  surface  of  the  water  around  H,  a  dia- 
phragm is  placed  in  the  centre  of  the  reservoir,  on  which  the  water 
impinges,  a  number  of  holes  being  bored  in  the  lower  portion  to 
equalize  the  level  on  each  side. 

Experiment.  When  water  flows  through  an  aperture  in  a  thin 
plate  the  amount  per  minute  is  much  less  than  that  given  by  the- 
ory, owing  to  the  contraction 
of  the  liquid  vein  immedi- 
ately after  leaving  the  orifice. 
The  ratio  of  the  two  is  called 
the  coefficient  of  efflux,  and 
the  whole  science  of  hydrau- 
lics is  based  on  this  constant. 
To  determine  it,  water  is  al- 
lowed to  flow  from  A  under 
a  given  head  through  an  ori- 
fice E,  and  the  quantity  meas- 
ured by  the  scale  attached 
to  H.  Place  one  of  the  cir- 
cular orifices  in  E,  and  meas- 
ure its  height  by  bringing  the  water  just  on  a  level  with  it,  and 
using  the  hook  gauge.  This  is  done  as  is  described  in  Ex- 
periment 13,  by  bringing  the  point  of  the  hook  just  to  the  surface 
of  the  liquid,  so  as  slightly  to  distort  the  image  of  outside  objects, 
and  reading  the  position  by  the  scale.  Close  E  with  the  rod  I£ 
and  open  the  valve  (7,  first  raising  the  funnel  F  nearly  to  the  top 


Fig.  41. 


96  EFFLUX    OF    LIQUIDS. 

of  the 'reservoir.  When  the  water  begins  to  escape  over  the  edge 
of  the  funnel  close  the  cock,  and  read  very  carefully  the  level  by 
the  gauge.  Read  also  the  height  of  the  liquid  in  _B,  which  should 
be  nearly  empty.  At  the  beginning  of  a  minute  open  E  by  re- 
moving the  rod  J£,  when  the  water  will  begin  to  flow  into  £  in  a 
clear  transparent  steam,  marked,  when  the  aperture  is  not  circular, 
by  alternate  swellings  and  contractions.  As  the  liquid  will  at  once 
descend  in  A,  the  valve  C  should  be  opened  at  the  same  time,  and 
adjusted  so  that  the  water  shall  slowly  trickle  over  the  edge  of  the 
funnel,  or  outlet  tube,  or  the  latter  may  be  dispensed  with,  and  the 
surface  kept  just  at  the  point  of  the  hook.  When  IB  is  nearly  full, 
which  should  take  at  least  five  minutes,  close  E  and  note  the  time. 
It  is  best  to  make  this  come  at  the  end  of  a  minute.  Now  read 
the  height  of  the  water  in  _#,  empty  it,  and  repeat  to  see  if  the 
same  results  are  obtained  twice  in  succession.  Make  the  ex- 
periment again  with  other  pressures,  also  changing  the  orifices. 

To  reduce  the  scale-readings  of  H  to  cubic  inches,  the  reservoir 
3  must  next  be  calibrated.  If  nearly  rectangular,  a  direct  meas- 
urement will  give  its  horizontal  cross-section,  but  if  the  sides  are 
at  all  curved  it  is  safer  to  use  some  other  method.  A  plan  much 
used  in  practice  is  to  mount  it  on  a  platform  scale  and  weigh  it 
when  empty,  and  when  filled  with  water  to  various  heights,  and 
reduce  the  weight  of  the  water  in  each  case  to  cubic  inches,  by 
dividing  by  .03614,  the  weight  in  pounds  of  one  cubic  inch  of  wa- 
ter. A  curve  should  then  be  constructed,  in  which  ordinates  rep- 
resent the  scale-readings  and  abscissas  the  volumes.  If  greater 
accuracy  is  required,  the  tenth  of  a  cubic  foot  used  in  Experiment 
19  should  be  employed.  A  T  is  placed  between  its  valve  and  the 
glass,  the  branch  of  which  is  connected  with  the  hydrant  by  a 
rubber  tube.  It  is  then  hung  over  the  reservoir  J5,  as  in  the  fig- 
ure. To  use  it,  admit  water  until  it  is  filled  to  the  top  of  the  hook 
in  its  upper  end.  Shut  off  the  water,  and  open  the  valve  below. 
When  the  water  level  has  reached  the  lower  point,  close  the  valve 
and  read  the  gauge  in  J?,  thus  taking  a  series  of  readings  which  will 
correspond  to  intervals  of  precisely  one  tenth  of  a  cubic  foot.  In 
this  case  it  is  best  to  construct  a  residual  curve  to  show  more 
clearly  the  irregularities  in  form  of  the  reservoir. 

The  area  of  the  orifices  must  next  be  measured  with  a  fine  scale, 


JETS    OF    WATER.  97 

reading  to  tenths  of  a  division  by  the  eye,  or  if  greater  accuracy 
is  required,  using  the  dividing  engine. 

Finally,  to  compute  the  theoretical  flow,  we  have  the  following 
data.  By  the  theorem  of  Torricelli  the  velocity  =  JZgh,  in 
which  g  —  32.2  ft.,  or  the  acceleration  of  gravity,  and  h  is  the 
height  of  the  liquid  above  the  centre  of  pressure  of  the  orifice.  This 
equals  the  difference  in  the  two  readings  of  the  hook  gauge  in  A, 
before  and  after  the  experiment,  correcting  for  the  positio.n  of  the 
centre  of  pressure,  which  will  sensibly  coincide  with  the  centre  of 
gravity  of  the  orifice.  Thus  with  a  circular  orifice  •  one-half  its 
diameter  must  be  subtracted.  A  stream  of  water  will  then  flow 
out  having  a  volume  equal  to  that  of  a  prism  with  cross-section  s 
equal  to  that  of  the  orifice,  and  a  length  v  for  each  second,  or  in 
t  seconds,  the  observed  time,  the  volume  V  should  be  stv  = 
st+/%gh.  The  observed  volume  is  obtained  directly  from  the  cali- 
bration of  7?,  of  which  either  a  curve  or  a  table  should  be  fur- 
nished. This  quantity  divided  by  "Fgives  m,  the  coefficient  of  efflux. 

49.    JETS  OP  WATER. 

Apparatus.  A  cylindrical  brass  tube  is  used  as  an  orifice,  and 
is  mounted  at  a  height  of  three  or  four  feet  from  the  floor,  with  a 
hinge  and  graduated  circle,  so  that  it  can  be  set  at  any  given 
angle.  A  deal  rod  divided  into  inches  is  attached  to  it  to  measure 
the  range,  and  the  whole  is  connected  with  the  hydrant  by  a  rub- 
ber tube  and  valve,  so  that  water  may  flow  through  it  at  any  re- 
quired velocity.  The  water  is  collected  as  it  escapes  in  a  large 
vessel,  which  is  weighed  in  a  spring  balance  before  and  after  the 
experiment,  and  thus  the  amount  of  water  determined.  A  second 
scale  of  inches  is  also  required  to  measure  the  vertical  descent  of 
the  curve. 

Experiment.  Almost  all  the  laws  of  projectiles  may  be  proved 
by  this  apparatus.  1st.  The  form  of  the  jet  is  a  parabola.  Set  the 
tube  horizontal,  and  allow  the  water  to  flow  through  it,  with  such 
a  velocity  that  in  moving  three  feet  horizontally  it  will  descend 
about  the  same  distance.  Take  care  that  this  velocity  is  un- 
changed during  the  experiment  by  noticing  that  the  horizontal 
range  remains  the  same.  Now  measure  the  vertical  fall  of  the  jet 
for  every  two  inches  on  the  horizontal  scale,  and  construct  a  curve 
with  these  distances  as  coordinates.  Next,  to  measure  the  veloc- 
7 


98  RESISTANCE    OF    PIPES. 

ity,  allow  the  water  to  flow  into  the  vessel  for  one  minute,  and 
weigh  it.  The  weight  in  grammes  equals  the  number  of  cubic 
centimetres,  and  this  divided  by  the  area  of  the  orifice  (found  by 
measuring  the  diameter  of  the  tube),  gives  the  velocity  of  the 
water  per  minute.  Divide  this  by  60,  for  the  velocity  per  second, 
and  construct  the  parabola  given  by  theory,  in  which  x  =  vt,  and 

at*        gx2 
y  =  -Q-  =  Q-g,  and  the  acceleration  of  gravity  g  =  386  inches. 

Great  care  must  be  taken  to  reduce  all  these  quantities  to  the  same 
measure,  as-  inches  or  metres,  several  different  units  being  pur- 
posely employed  in  these  measurements.  Repeat  the  latter  part 
of  this  experiment  with  three  or  four  different  velocities,  and  see 
if  for  a  given  value  of  y,  x  is  proportional  to  v. 

2d.     The  horizontal  range  for  a  velocity  v,  and  angle  of  projec- 

v2 

tion  «,  equals  —  sin  2a.     Prove  this  by  measuring  the  range  for 
y 

every  5°  from  0°  to  90°.  Evidently  the  maximum  is  when  x  = 
45°.  In  a  similar  manner  we  may  prove  that  the  maximum  range 
on  an  inclined  plane  is  attained  when  the  direction  of  the  jet 
bisects  the  angle  between  it  and  the  vertical,  and  again,  that  the 
curve  of  safety  or  envelope'  to  all  the  parabolas  formed  with  a 
given  velocity  when  the  jet  is  turned  in  different  directions,  is  a 
parabola,  with  the  orifice  for  a  focus. 

50.     RESISTANCE  OF  PIPES. 

Apparatus.  A  f"  brass  tube  six  feet  in  length  has  five  holes 
drilled  in  it  at  intervals  of  exactly  a  foot,  taking  care  that  no  burr 
or  roughness  remains  on  the  inside.  Short  pieces  of  brass  tubing 
are  soldered  on  over  them,  and  long  glass  tubes  are  attached  by 
pieces  of  rubber  hose.  The  whole  is  mounted  on  a  stand,  so 
that  the  brass  pipe  is  horizontal,  and  the  glass  tubes  vertical  and 
a  foot  apart.  Each  tube  is  graduated,  or  has  a  paper  scale  at- 
tached, .to  show  the  height  at  which  the  water  stands  in  it.  Wa- 
ter may  be  passed  through  the  brass  pipe  at  different  velocities  by 
connecting  it  with  the  hydrant,  and  regulating  the  flow  by  the  fau- 
cet. To  keep  the  pressure  regular,  it  is  better  to  connect  with  a 
separate  reservoir,  and  to  measure  the  velocity,  the  water  may  be 
received  in  a  large  graduated  vessel. 

Experiment.  When  water  flows  through  the  brass  pipe  it  will 
rise  in  the  glass  tubes  owing  to  the  friction,  and  the  latter  may  be 


FLOW    OF    LIQUIDS    THROUGH    SMALL    ORIFICES.  99 

very  accurately  measured  by  the  height  of  the  liquid.  On  trying 
the  experiment  it  will  be  noticed  that  the  top  of  the  liquid  col- 
umns lie  very  nearly  in  a  straight  line,  passing  through  the  open 
end  of  the  pipe,  where  of  course  the  pressure  is  zero.  The  exact 
pressure  should  be  measured  by  the  attached  scale,  and  observa- 
tions of  all  of  them  taken  for  several  different  heights.  A  second 
series  of  experiments  should  also  be  made  to  determine  the  veloc- 
ity corresponding  to  these  heights.  In  this  case  the  escaping 
liquid  is  received  in  the  graduated  vessel  for  a  known  time,  or  the 
time  required  to  fill  it  is  noted,  and  from  this,  knowing  the  volume 
and  cross-section  of  the  pipe,  the  velocity  is  readily  determined. 
The  results  should  be  represented  by  curves,  first  making  abscissas 
distances,  and  ordinates  pressures,  and  secondly,  using  velocities 
as  abscissas,  and  the  heights  of  the  liquid  in  the  most  distant  tube 
for  ordinates.  From  these  curves  the  laws  and  coefficients  of 
liquid  friction  are  readily  determined.  i 

51.      FLOW  OF  LIQUIDS  THROUGH  SMALL  ORIFICES. 

Apparatus.  A  Mariotte's  flask  is  placed  about  three  feet  above 
the  table  and  a  rubber  tube  is  connected  with  its  outlet.  To  this 
is  fastened  a  brass  tube  with  a  perforated  screw  cap,  so  arranged 
that  small  circles  of  platinum  foil  may  be  inserted,  with  holes  of 
various  sizes.  A  vertical  scale  shows  the  height  of  the  orifice,  and 
a  balance  serves  to  measure  the  quantity  of  water  received. 

Experiment.  Fill  the  Mariotte's  flask  with  water.  For  this 
purpose  it  is  often  convenient  to  have  a  third  tube,  which  is  closed 
by  a  rubber  cap,  except  when  the  flask  is  to  be  filled.  It  is  then 
opened  to  allow  the  air  to  escape,  and  water  is  admitted  by  one  of 
the  other  tubes.  Raise  the  orifice  so  that  water  is  just  on  the 
point  of  flowing  out  of  it,  and  measure  its  height.  Insert  one  of 
the  platinum  diaphragms  and  lower  it,  so  that  the  water  shall  flow 
out  drop  by  drop.  Collect  what  escapes  during  a  minute,  and 
weigh  it.  Lower  the  orifice  and  repeat  at  intervals,  until  it  is  as 
low  as  possible.  Measure  also  at  the  point  where  the  drops  begin 
to  unite  into  a  continuous  stream.  For  all  lower  points  measure 
the  length  of  the  stream,  that  is,  the  distance  before  it  begins  to 
divide  into  drops. 

Compute  the  coefficient  of  efflux  by  means  of  the  usual  formula, 


100  CAPILLARITY. 

Y 

V  =  mstv  =  mst*/2gh,  hence  m  =        .~   »,  in  which  s  equals  the 

7T6?2 

cross  section  =  ^T,  calling  d  the  diameter  of  the  orifice,  t  —  the 

time  of  flow  =  60,  h  the  head,  or  the  reading  first  taken  minus 
that  corresponding  to  the  given  observation,  and  V  is  obtained 
from  the  weight,  remembering  that  1  gramme  of  water  =  1  cm3  = 
.061  inches.  Finally,  construct  a  curve  in  which  ordinates  rep- 
resent the  coefficients  m,  and  abscissas  the  heads  h. 

By  this  simple  apparatus,  interesting  results  could  be  obtained  by 
measuring  the  flow  of  various  liquids  with  different  pressures  and 
orifices.  Their  relative  viscosity  might  thus  be  compared. 

52.     CAPILLARITY. 

Apparatus.  In  Fig.  42,  A  is  a  tall  bell-glass  set  in  a  glass  jar 
IB  containing  water.  C  is  a  glass  tube  drawn  out  to  a  point  and 
Connected  with  A  by  a  rubber  tube ;  it  is  immersed  in  a  test  tube 
J>,  containing  the  liquid  to  be  tried.  A  may  be  filled  with  air  by 
blowing  through  the  bent  tube  E.  Paper  scales  divided  into  milli- 
metres are  attached  to  J3  and  D  to  measure  the  pressure,  and  D  is 
supported  in  such  a  way  that  it  may  be  raised  or  lowered  at  will. 

Experiment.  Draw  out  a  piece  of  glass  tubing  to  a  fine  point, 
break  off  a  small  piece  and  grind  the  end  flat  so  that  the  orifice 
shall  be  circular  and  smooth.  Connect 
it  as  at  (7,  by  a  rubber  tube,  with  the 
bell-glass  A^  and  fill  the  latter  with  air  by 
blowing  into  E.  Raise  the  test-tube  D 
containing  the  liquid  to  be  employed,  so 
that  the  air  escaping  from  C  shall  bubble 
up  through  it.  Soon  the  pressure  in  A  is 
so  far  diminished  that  it  becomes  insuffi- 
cient to  overcome  the  resistance  opposed  to  it,  the  flow  will  then 
stop,  and  the  top  of  the  liquid  in  C  w;ll  be  found  to  be  very  much 
curved.  Record  the  pressure  of  the  air  in  A  which  equals  the 
difference  in  level  of  the  water  within  and  without  it.  Call  it  A, 
and  call  hf  the  difference  in  level  within  and  without  C.  Repeat 
this  observation  several  times,  either  by  blowing  into  E,  or  by 
lowering  D  so  that  the  flow  shall  recommence.  Next  remove  the 
tube  from  the  liquid,  break  off  the  end,  and  stick  it  carefully  into 


PLATEAU'S  EXPEKIMENT.  101 

a  cork.  Grind  down  the  end  of  C  until  it  is  again  flat,  and  repeat 
until  observations  have  been  obtained  with  orifices  of  five  or  six 
different  sizes.  Now  place  the  cork  on  the  dividing  engine,  Ex- 
periment 21,  and  measure  the  diameter  of  each  of  the  ground  ends. 
This  may  be  obtained  with  great  accuracy  by  placing  the  axis  of 
the  tube  vertical  so  as  to  look  down  through  it. 

Let  s  be  the  specific  gravity  of  the  liquid  and  x  the  height  to 
which  it  would  tend  to  rise  in  the  tube,  if  the  bore  were  the  same 
throughout  as  at  the  end.  The  pressure  due  to  this  force  will 
then  be  sx,  and  in  the  same  way  the  pressure  due  to  the  column 
hr  will  be  sh'.  Both  of  these  pressures  will  be  in  equilibrium  with 
the  force  h  of  the  water  in  JB.  In  other  words,  h  =  sx  -\-  shf,  or 

x  _  _        L_?  froni  which  x  may  be  calculated  in  the  various  cases. 
s 

If  the  liquid  in  D  is  water,  s  =  1,  and  x  =  h  —  hf.  This  method 
of  studying  capillarity  was  first  proposed  by  M.  Simon,  who,  how- 
ever, found  that  his  results  did  not  agree  with  those  obtained  by 
direct  measurement.  It  has,  however,  the  great  advantage  that 
the  diameter  may  be  obtained  with  accuracy,  even  with  very  mi- 
nute tubes,  and  the  latter  being  heated  to  redness  are  rendered 
chemically  clean. 

53.    PLATEAU'S  EXPERIMENT. 

Apparatus.  Some  of  Plateau's  soap-bubble  mixture,  formed  by 
mixing  pure  oleate  of  soda  with  30  parts  of  water,  and  adding  two 
thirds  its  bulk  of  glycerine.  The  oleate  is  made  of  olive  oil  and 
soda,  which  is  then  filtered.  Common  soap  may  however  be  used. 
Wires  are  bent  into  the  following  forms  and  soldered  at  the  cor- 
ners. A  tetrahedron  with  a  single  wire  as  a  handle,  a,  cube,  a  cir- 
cle, two  triangles  hinged  along  one  side,  and  two  squares,  made  in 
the  same  way,  also  a  small  vertical  stand  arranged  so  that  two 
circles  may  be  placed  on  it  at  any  height.  To  measure  the  figures 
obtained,  an  upright  (7,  Fig.  43,  is  attached  to  the  table  to  support 
a  sheet  of  paper  and  at  a  distance  of  about  two  feet  is  a  second 
support,  A,  in  which  is  a  small  hole  to  look  through.  A  third 
stand,  B,  serves  to  hold  the  wire  figures  in  any  desired  position. 

Experiment.  Dip  the  tetrahedron  into  the  liquid,  and  on  draw- 
ing it  out,  films  will  be  found  extending  from  each  of  the  six 
edges,  and  meeting  in  the  centre.  This  point  is  a  fourth  of  the 
distance  from  each  face  to  the  opposite  angle.  Attach  the  tetrahe- 


102  PLATEAU'S  EXPERIMENT. 

dron  to  B,  so  that  one  face  shall  be  nearly  horizontal,  and  one 
edge  perpendicular  to  the  line  through  ABC.  On  looking 
through  A  it  is  projected  as  a  triangle  on  (7.  Move  _Z?,  if  nec- 
essary, so  that  its  lower  face  shall  be  projected  as  a  straight  line. 
Attach  a  piece  of  paper  to  C 
and  mark  on  it  the  corners  of  the 
tetrahedron,  also  the  intersection  of 
the  films.  Measure  on  the  paper  the 
distance  of  this  point  from  the  top 


and   bottom  of  the   figure.     Their 

ratio  should  be  one  to  three.  Turn  the  tetrahedron  around,  and 
repeat  the  measurement  with  one  of  the  other  sides.  It  should 
be  the  same  for  all. 

A  general  law  of  these  films  is  that  they  are  always  subjected  to 
tension  and  continually  tend  to  contract,  owing  to  the  molecular 
attraction  of  the  particles.  This  may  be  shown  in  various  ways. 
Attach  a  loop  of  the  finest  silk  thread  to  the  circle  of  wire.  Dip  it 
in  the  liquid,  and  a  film  will  be  obtained  in  which  the  loop  will  float, 
irregular  in  shape  and  in  any  position.  Break  the  film  inside  the 
loop,  and  instantly  by  the  contraction  of  the  film  around  it,  it  will 
be  drawn  out  into  a  perfect  circle,  leaving  of  course  a  hole  in  the 
centre.  Inclining  the  circle  from  side  to  side  the  loop  moves 
freely  over  the  film,  presenting  the  curious  appearance  of  a  sheet 
of  liquid  containing  a  moveable  hole. 

Immerse  the  tetrahedron  again  in  the  liquid.  The  six  films  pul- 
ling equally  in  'opposite  directions,  hold  the  centre  point  in  equi- 
librium. Now  break  one  of  the  films,  and  the  remainder  con- 
tracts, forming  a  curious  curved  surface  drawn  towards  one  side 
by  a  single  plane  film.  On  breaking  this  second  film,  the  surfaces 
again  contract  and  form  the  warped  surface  known  as  the  hyper- 
bolic paraboloid. 

Immersing  the  cube  in  the  same  way  twelve  plane  surfaces  are 
obtained,  meeting  in  a  small  square  in  the  centre.  This  square 
may  be  parallel  to  either  face,  aud  may  be  made  to  alter  its  posi- 
tion by  gently  blowing,  so  as  in  appearance  to  split  it.  See  how 
many  different  figures  can  be  obtained  by  breaking  one  or  more 
films,  and  draw  them  in  your  note  book.  The  whole  number  is 
twelve,  not  including  a  single  plane  attached  to  one  face  only.  If 


PNEUMATICS.  103 

the  films  attached  to  two  opposite  parallel  sides  are  broken,  a  plane 
is  obtained  supported  between  two  curved  surfaces,  the  intersec- 
tions being  curved  lines.  Draw  these  lines  by  attaching  the  cube 
to  B  and  see  if  they  are  hyperbolas.  Another  curious  effect  is  ob- 
tained by  blowing  a  small  bubble  and  attaching  it  to  the  centre 
square,  when  it  assumes  a  cubical  form  with  curved  sides ;  in  the 
same  way  a  four-sided  bubble  may  be  formed  with  the  tetrahe- 
dron. Similar  figures  may  be  obtained  with  an  octahedron, 
or  other  figures,  but  they  are  more  complex. 

On  dipping  the  two  triangles  into  the  liquid  a  film  forms  over 
both,  and  on  increasing  the  angle  between  them  a  single  plane  film 
is  found  attached  to  their  common  side,  which  is  split  as  they  sepa- 
rate. Breaking  this  film  the  curve  springs  back  as  before,  forming 
a  very  beautiful  hyperbolic  paraboloid.  This  is  probably  the  best 
way  of  producing  this  warped  surface,  and  its  properties  are  well 
shown  by  it.  Varying  the  angle  between  the  triangles,  its  form,  or, 
more  strictly,  its  parameter  may  be  altered  at  will.  Make  the 
angle  between  the  triangles  about  30°,  and  draw  the  curve  of  in- 
tersection of  the  plane  film  with  the  other,  also  a  section  through 
the  centre  at  right  angles  to  it.  Try  and  determine  the  form  of 
the  first  of  these  curves,  and  see  if  it  is  a  circle,  parabola  or  hyper- 
bola. Now  break  the  film  and  draw  the  enveloping  curves  on  the 
same  sheet  as  before,  to  show  how  the  films  have  contracted. 
Do  the  same  with  the  jointed  squares.  Place  the  two  circles 
on  their  stand  near  together,  blow  a  bubble  and  lay  it  on  them. 
Then  draw  them  apart,  and  a  hyperboloid  of  revolution  of  one 
nappe  will  be  obtained. 

54.    PNEUMATICS. 

Apparatus.  The  object  of  this  experiment  is  to  familiarize  the 
student  with  the  ordinary  lecture-room  apparatus  in  pneumatics, 
and  is  therefore  chiefly  of  value  to  those  who  propose  to  adopt 
teaching  as  a  profession.  The  apparatus  needed  will  depend  on 
the  objects  of  each  student,  but  may  be  made  to  include  almost  all 
the  instruments  used  in  a  full  course  of  lectures  on  this  branch  of 
physics.  The  following  description,  however,  applies  only  to  such 
experiments  as  could  properly  be  introduced  in  any  common 
school.  The  most  important  instrument  is  of  course  the  air  pump, 
which  need  not  be  of  large  size,  or  (for  most  of  these  experiments) 
capable  of  producing  a  very  high  degree  of  exhaustion.  The 


104  PNEUMATICS. 

other  apparatus  needed  is  best  determined  from  the  following  list 
of  experiments,  which  may  be  varied  almost  indefinitely. 

Experiment.  Place  a  receiver  on  the  pump-plate,  taking  care 
that  no  dust  or  grit  is  retained  under  the  edge,  which  should  be 
freely  supplied  with  sperm  oil,  or  tallow,  to  ensure  contact.  Open 
communication  between  the  pump  and  receiver,  and  close  that 
leading  to  the  outer  air.  Exhaust,  by  working  the  handle  of  the 
pump,  and  see  if  any  leakage  takes  place  around  the  bottom  of  the 
receiver,  in  which  case  air  bubbles  will  be  seen  forcing  their  way 
through  the  oil.  The  greatest  trouble  in  using  the  air-pump  is  to 
make  this  joint  tight,  especially  if  the  plate  or  receiver  is  not 
ground  perfectly  true.  When  the  exhaustion  is  nearly  complete 
the  pump  handle  will  work  freely,  until  the  very  end  of  the  stroke, 
when  a  slight  hissing  will  be  heard,  due  to  the  expulsion  of  the 
remaining  air.  For  this  reason  the  piston  must  be  moved  until  it 
strikes  the  end  of  the  cylinder  each  time,  and  the  strokes  must  be 
taken  steadily,  and  not  too  fast.  When  the  air  is  removed  the 
exterior  pressure  becomes  so  great  that  it  is  impossible  to  move  the 
receiver  without  breaking  the  glass.  On  opening  communication 
with  the  outer  air,  the  latter  rushes  in,  and  the  receiver  is  easily 
removed.  To  determine  the  degree  of  exhaustion,  a  syphon 
vacuum  gauge  may  be  employed.  This  consists  of  a  bent  glass  tube 
like  a  syphon  barometer,  with  the  closed  end  only  about  half  a 
foot  in  length,  and  containing  mercury,  which  of  course  rises  to  the 
top.  Place  it  under  a  receiver  and  exhaust,  when  it  will  be  found 
that  as  soon  as  the  pressure  inside  is  reduced  to  less  than  six  inches 
the  mercury  begins  to  fall,  until  in  a  perfect  vacuum  it  would  stand 
at  the  same  height  in  both  branches  of  the  tube.  Read  the  differ- 
ence in  level,  which  in  a  common  pump  should  not  exceed  two  or 
three  millimetres.  If  a  barometer  gauge,  or  long  tube  dipping  in 
mercury,  is  attached  to  the  pump,  subtract  its  reading  from  that  of 
the  standard  barometer,  and  the  difference  should  equal  that  of  the 
syphon  gauge.  Place  a  beaker  of  water  on  the  pump-plate  with  a 
bolt  head  (or  tube  with  a  bulb  blown  at  one  end)  in  it,  cover  with 
a  receiver  and  exhaust  slowly.  The  air  will  now  bubble  up 
through  the  water,  owing  to  its  tendency  to  expand  when  the 
outer  pressure  is  removed.  If  the  pump  is  a  very  nice  one,  this 


PNEUMATICS.  105 

experiment,  and  others  requiring  water,  should  be  omitted,  as  the 
vapor  may  rust  the  interior  of  the  pump.  On  readmitting  the  air 
the  water  will  rush  up  into  the  bolt-head  until  but  a  small  bubble 
of  air  remains.  The  ratio  of  the  volume  of  this  bubble  to  the 
whole  interior  of  the  bolt-head,  shows  the  degree  of  exhaustion. 
When  nearly  all  the  pressure  of  the  air  is  removed  from  the  sur- 
face, the  water  bubbles  make  their  appearance  in  it,  due  to  the 
dissolved  air.  Carrying  the  exhaustion  still  farther,  vapor  begins 
to  be  formed  so  rapidly  that  the  water  enters  into  ebullition.  This 
effect  is  more  easily  obtained  if  the  water  is  somewhat  warm. 
Select  two  tubes  about  three  feet  long,  and  closed  at  one  end,  fill 
one,  j5,  with  mercury  (Experiment  58),  the  other,  ^4,  with  air,  and 
dip  both  into  a  small  vessel  containing  mercury.  Cover  them  with 
a  tall  receiver  and  exhaust.  The  mercury  will  descend  in  13  until 
nearly  on  a  level  with  that  in  the  cistern,  the  air  meanwhile  escap- 
ing from  A  in  bubbles.  Readmit  the  air  and  the  mercury  will  rise 
in  both  tubes,  that  in  A  being  the  lowest.  Any  leakage  in  the 
pump  is  well  shown  in  these  experiments,  as  it  will  cause  the  li- 
quid to  begin  to  rise  slowly  as  soon  as  the  pumping  stops.  To  see 
if  the  leak  is  in  the  pump,  or  under  the  receiver,  close  the  connec- 
tion between  them  when  leaks  in  the  latter  only  will  be  percepti- 
ble. The  great  pressure  of  the  air  may  be  shown  in  various  ways. 
Thus  the  palm-glass  is  a  cylindrical  vessel  open  at  both  ends, 
which  is  placed  on  the  pump-plate  and  closed  above  by  the  hand ; 
after  exhaustion  the  latter  is  removed  only  with  difficulty.  Re- 
placing the  hand  by  a  sheet  of  rubber,  a  single  stroke  of  the  pump 
will  draw  it  strongly  inwards,  and  in  the  same  way  a  tightly 
stretched  bladder  may  be  made  to  burst  with  a  loud  report.  In 
the  upward  pressure  apparatus  the  air,  being  withdrawn  above  a 
piston,  the  latter,  with  a  heavy  weight  attached,  is  raised  by  the 
pressure  of  the  air  below.  The  Magdeburg  hemispheres  consist 
of  two  brass  hemispheres,  accurately  ground  together,  which  re- 
quire a  great  force  to  separate  them  when  the  air  is  with  drawn 
from  the  interior.  Great  care  is  needed  in  handling  this  apparatus 
as  a  slight  blow  will  bend  the  brass  sufficiently  to  cause  leakage. 
Bursting  squares  are  sealed  rectangular  vessels  of  glass,  which 
explode  when  placed  under  an  exhausted  receiver.  To  prevent 
injury  they  should  be  covered  with  wire  gauze,  and  the  orifice 


106  PNEUMATICS. 

leading  to  the  purnp,  protected  by  a  brass  cap  and  valve.  The 
porosity  of  wood  may  be  shown  by  the  mercury  funnel,  in  which 
mercury  is  driven  lengthwise  through  a  piece  of  wood  which 
passes  through  the  top  of  a '  receiver.  If  a  piece  of  wood  held 
under  water  is  covered  with  a  receiver,  and  the  air  exhausted,  tor- 
rents of  bubbles  imprisoned  in  its  pores  will  pour  from  it.  Now 
on  admitting  the  air  the  water  enters  the  wood,  which  becomes 
water-logged,  and  no  longer  floats.  The  revolving  jet  is  a  bent 
brass  tube,  like  a  Barker's  Mill,  which  when  placed  under  a  receiver 
turns  rapidly  in  one  direction  when  the  air  is  exhausted,  and  in 
the  other  when  it  is  readmitted.  The  effect  of  the  resistance  of 
the  air  is  shown  by  two  fan  wheels  with  vanes  set  flatwise  and 
edgewise,  respectively.  If  set  in  motion,  the  former  stops  first  in 
air,  but  both  revolve  for  nearly  the  same  time  in  a  vacuum.  The 
same  effect  may  be  shown  by  a  feather  and  guinea  placed  in 
a  long  glass  tube  from  which  the  air  is  removed.  They  then 
fall  with  nearly  equal  velocity  from  end  to  end.  An  import- 
ant experiment  is  the  proof  of  the  weight  of  the  air.  A  glass 
sphere  is  weighed  when  full  of  air  and  when  exhausted,  and  the 
difference  gives  approximately  the  required  weight.  The  exact 
weight  is  obtained  only  by  an  accurate  correction  for  temperature, 
pressure  and  moisture.  The  two  following  experiments,  though 
properly  belonging  to  other  branches  of  physics,  are  inserted  here 
for  convenience.  Both  require  a  very  high  degree  of  exhaustion. 
If  a  bell  is  rung  in  a  vacuum,  no  sound  is  heard.  An  electric  bell 
is  most  convenient  for  this  experiment.  It  should  be  carefully 
supported,  so  that  the  sound  shall  not  be  transmitted  directly  to 
the  pump  plate.  For  this  purpose  it  is  sometimes  hung  by  threads ; 
a  rubber  support  is  also  recommended.  The  experiment  is  gen- 
erally more  successful,  if  after  exhaustion  hydrogen  gas  is  ad- 
mitted, and  the  exhaustion  repeated.  It  is,  however,  almost  im- 
possible to  destroy  all  sound.  The  latent  heat  of  aqueous  vapor  is 
well  shown  by  the  experiment  of  freezing  water  in  vacuo.  A 
shallow  pan  of  concentrated  sulphuric  acid  l  is  placed  on  the  pump 
plate,  and  on  this  a  wire  triangle  which  supports  a  flat  metallic 

1  In  all  cases  where  sulphuric  acid  is  used  to  absorb  moisture  in  the  presence  of  metallic 
surfaces,  it  should  be  freed  from  nitric  fumes  by  boiling  it  for  some  time  with  sulphate 
of  ammonia. 


107 

dish  holding  the  water  to  be  frozen.  The  whole  is  covered  with  a 
small  receiver,  and  exhausted  quickly.  On  removing  the  pressure 
from  its  surface  the  water  is  rapidly  converted  into  vapor,  which  is 
absorbed  by  the  sulphuric  acid  as  fast  as  formed ;  the  action  there- 
fore continues,  the  latent  heat  being  obtained  at  the  expense  of  the 
water,  which  accordingly  cools  until  it  is  converted  into  ice.  By 
substituting  for  water  more  volatile  substances,  as  a  mixture  of 
solid  carbonic  acid  and  ether,  and  adding  protoxide  of  nitrogen,  the 
most  intense  cold  yet  observed  is  attained.  In  the  best  pumps 
water  may  be  frozen  by  its  own  evaporation,  without  employing 
acid  to  absorb  its  vapor. 

55.    MAKIOTTE'S  LAW. 

Apparatus.  A  modification  of  Regnault's  apparatus  may  be 
made  chiefly  with  steam  fittings,  as  shown  in  Fig.  44.  C  is  a  tall 
mercury  gauge  formed  of  glass  tubes,  connected  together  by  a 
steam  pipe  coupling  with  red  sealing-wax.  If  very  high,  all  the 
joints  must  be  made  like  those  of  Regnault's  gauge,  but  this  is 
unnecessary  for  pressures  below  one  hundred  pounds.  A  and  B 
are  two  similar  tubes  about  three  feet  long,  closed  above  by  "pet- 
cocks,"  and  attached  below  by  "  unions,"  so  that  they  may  be  easily 
removed.  E  is  a  reservoir  made  of  3  inch  pipe  with  caps,  to  hold 
the  mercury,  and  with  an  £"  valve  below,  so  that  it  may  be 
emptied  if  necessary.  It  is  filled  by  removing  the  plug  in  the  T 
at  G.  I  is  a  small  force-pump  such  as  is  used  in  testing  gauges, 
by  which  water  may  be  drawn  from  the  reservoir  K,  and  forced 
into  E.  The  water  is  allowed  to  flow  back  by  opening  the  valve 
H.  Remove  the  plug  6r,  and  pour  into  E  enough  mercury  to  fill 
A,  £  and  C.  Work  the  pump  slowly  until  E  is  full  of  water. 
Then  close  6r  and  expel  any  air  that  may  remain  by  working  I 
and  opening  H  alternately,  until  no  air  bubbles  rise  up  through 
the  water  in  K.  Scales  are  attached  to  A,  IB  and  (7,  and  the  first 
two  should  be  carefully  calibrated  (Experiment  10).  IB  may  be 
permanently  filled  with  dry  carbonic  acid,  or  other  gas. 

Experiment.  A  must  first  be  filled  with  dry  air.  For  this  pur- 
pose connect"  it  above  with  a  U-tube  containing  chloride  of  lime, 
open  the  pet-cock  and  pump  up  the  mercury  nearly  to  the  top, 
thus  forcing  out  the  air.  Open  II  a  very  little,  and  let  the  mer- 
cury slowly  descend.  The  air  is  thus  drawn  into  A,  first  being 
dried  by  passing  over  the  lime.  Repeat  several  times  to  expel  all 
the  moisture  that  may  remain,  and  finally,  when  full  of  dry  air, 
close  the  pet-cock.  Read  very  carefully  the  height  of  the  mer- 


108 


MARIOTTE'S  LAW. 


Fig.  44. 


cury  A,  B  and  C,  and  record  in  three  columns.     Work  the  pump 
a  few  times,  and  take  readings  at  intervals  of  about  10  inches, 

until  the  mercury  has  nearly 
reached  the  top  of  A  C.  Note 
the  height  of  the  barometer  and 
the  readings  of  C  and  A,  when 
the  latter  is  open  to  the  atmos- 
phere, also  the  height  of  the 
mercury  in  A  when  standing  at 
the  same  level  as  in  C.  Write 
in  the  4th  and  5th  columns  the 
pressure  in  each  case,  found  by 
adding  to  the  height  of  the  ba- 
rometer the  difference  in  level 
of  the  mercury,  columns.  In  the 
6th  and  7th  columns  give  the 
volume  of  the  gas  in  each  case, 
deduced  from  the  table  of  calibration  of  A  and  B.  Next  take 
the  product  of  the  pressure  and  volume,  which  would  be  constant 
if  Mariotte's  law  were  correct,  or  the  volumes  inversely  as  the 
pressures.  Finally,  construct  curves  for  the  two  gases,  making 
abscissas  represent  these  products,  and  ordinates  pressures.  These 
results  will  be  only  approximate,  owing  to  the  change  of  tempera- 
ture the  gas  undergoes  when  rarefied  or  condensed.  To  diminish 
this  error  an  interval  should  be  allowed  for  the  gas  to  attain  the 
temperature  of  the  air  of  the  room,  or  better,  A  and  B  should 
be  surrounded  with  a  water  jacket,  the  temperature  carefully 
noted,  and  a  correction  applied. 

Much  greater  accuracy  is  attained  by  the  following  arrange- 
ment. A  third  tube  is  employed,  in  the  upper  part  of  which  two 
platinum  wires  pointing  upwards  are  inserted,  the  volume  above 
them  being  determined  very  accurately  by  inverting,  and  weigh- 
ing the  mercury  required  to  fill  it.  This  portion  of  the  tube  is 
then  enclosed  in  a  larger  one,  through  which  water  is  kept  circu- 
lating, and  its  temperature  noted  by  a  thermometer.  Fill  the 
tube  in  the  usual  way  with  dry  gas,  then  condense  it  until  the 
mercury  is  just  on  a  level  with  the  upper  platinum  point.  The 
mercury  in  C  should  now  stand  near  the  top  of  the  tube.  Open 


GAS-HOLDER. 


109 


IT,  and  let  the  pressure  diminish  until  the  mercury  in  the  third 
tube  is  exactly  on  a  level  with  the  lower  platinum  point.  Record 
the  pressure  in  each  case.  To  bring  the  mercury  to  the  exact 
level,  raise  it  by  the  pump  and  lower  by  opening  If  until  the 
point  is  just  perceptible  by  the  slight  distortion  it  produces  in  the 
image  of  objects  reflected  in  the  surface  of  the  mercury.  Let  the 
pressure  diminish  to  a  few  inches  of  mercury,  let  out  a  little  gas 
and  repeat.  The  law  may  be  tested  for  pressures  less  than  one 
atmosphere  by  merely  drawing  off  the  mercury  in  C  until  it  stands 
below  that  in  the  other  tube.  The  ratio  of  the  volumes  being 
constant  in  this  experiment,  the  ratio  of  the  pressures  would  be 
its  reciprocal,  if  Mariotte's  law  were  correct.  The  deviation  may 
be  shown  by  a  curve  in  which  abscissas  represent  the  smaller 
pressure,  and  ordinates  the  ratio  of  the  two  pressures. 

56.    GAS-HOLDEK. 

Apparatus.  A  good  gas-holder  containing  three,  or  better,  five 
cubic  feet,  with  scale  attached,  the  bell  properly  counterpoised, 
and  most  important  of  all,  the  friction  reduced  to  a  minimum.  To 
calibrate  it,  the  standard  tenth  of  a  cubic  foot  of  Experiment  19  is 
employed,  and  to  measure  the  pressure  some  very  delicate  form  of 
gauge  should  be  provided. 

Experiment.  The  gas-holder  consists  of  a  large  bell,  AB,  Fig. 
45,  suspended  in  a  circular  trough 
of  water,  and  counterpoised  by  the 
weight  F  attached  to  a  cord  passing 
over  the  pulley  D.  A  curved  piece  of 
metal  E,  called  the  cycloid,  is  attached 
to  this  pulley,  and  carries  a  second 
weight  G-,  which  acts  at  a  longer  and 
longer  arm  as  the  bell  rises.  It  thus 
compensates  for  the  diminution  of 
weight  of  the  bell  when  submerged? 
nd  renders  the  pressure  nearly  the 
same  whether  the  holder  is  full  or 
empty.  The  proper  form  for  E  is  the 
involute,  a  curve  in  which  the  perpen- 
dicular on  the  tangent  is  proportional 
to  the  angle  described  by  the  radius 


Fig.  45. 


110  GAS-HOLDER. 

vector.  The  gas  is  drawn  out  by  a  large  tube  opening  into  the 
bottom  of  the  holder  at  j5,  and  covered  at  K  by  a  cap  with  a 
water  seal.  A  large  tube  with  a  stopcock  H,  also  opens'  out  of  it, 
through  which  the  gas  may  be  drawn,  or  if  preferred,  through 
a  small  tube  just  below  it.  To  fill  the  holder  with  air,  remove  IL 
and  press  down  on  F,  when  the  bell  will  rise  to  the  top ;  it  may 
be  kept  there  by  replacing  J^  and  closing  H.  If  gas  is  to  be 
used,  it  must  be  filled  through  H,  but  this  is  a  much  slower  process. 

A  variety  of  experiments  may  be  performed  with  this  apparatus. 
First,  test  the  holder.  Fill  the  bell  nearly  full  of  air,  depress 
it  a  little  by  the  hand,  let  it  return,  and  record  the  reading  of  the 
scale.  Then  raise  it,  let  it  descend,  and  again  read.  Repeat  sev- 
eral times,  and  the  difference  in  the  results  shows  the  greatest 
error  due  to  friction.  Do  the  same  with  other  parts  of  the  scale. 
Next,  calibrate  the  bell.  The  same  method  is  employed  as  in  Ex- 
periment 48,  only  the  air  is  collected  instead  of  the  water.  In  this 
case,  after  emptying  the  holder,  add  one  tenth  of  a  cubic  foot  of  air 
at  a  time,  and  read  the  scale  after  each  addition.  Repeat  drawing 
out  one  tenth  of  a  cubic  foot  from  the  holder  into  the  glass  stand- 
ard, and  see  if  the  readings  are  the  same  as  before.  The  great 
difficulty  in  this  experiment  is  the  change  of  volume  of  the  air 
due  to  changes  of  temperature.  As  the  bell  rises  from  the  water 
the  adhering  moisture  evaporates,  and  sometimes  lowers  its  tem- 
perature very  rapidly.  It  is,  however,  customary  to  assume  that 
the  air  is  saturated  with  moisture,  and  at  the  same  temperature  as 
the  water  with  which  it  is  in  contact. 

Next,  measure  the  pressure  for  different  parts  of  the  scale  to  see 
if  the  compensation  is  exact.  The  gauge  is  attached  to  a  small 
tube  just  below  If,  with  an  independent  outlet  from  the  bell.  To 
save  time  the  pressure  may  be  observed  after  adding  each  tenth 
of  a  foot  in  the  last  experiment.  Various  forms  of  gauges  may  be 
employed.  The  simplest  is  a  large  U-tube,  with  a  scale  attached 
to  each  branch.  The  pressure  may  thus  be  determined  within  a 
hundredth  of  an  inch.  For  greater  accuracy,  a  bell  glass  standing 
in  water  fnay  be  connected  with  the  holder,  and  the  difference  in 
level  of  the  water  within  and  without  it  gives  the  pressure.  By 
using  two  hook  gauges  for  this  purpose  great  accuracy  may  be 
attained.  A  method  in  common  use  is  in  principle  similar  to  the 


GAS-METERS.  Ill 

wheel  barometer.  A  small  bell  is  connected  with  the  interior  of 
the  holder,  and  its  rise  and  fall  is  measured  by  a  cord  passing  over 
a  pulley  which  moves  a  pointer  over  a  graduated  circle.  If  the 
pressure  increases  as  the  holder  rises,  the  weight  G  should  be 
increased,  and  the  contrary  if  it  diminishes.  The  pressure  to  which 
the  gas  is  subjected  is  varied  by  changing  the  weight  F.  Prove 
this,  and  determine  the  law.  Do  the  same  for  different  parts  of 
the  scale.  To  test  the  above  work,  fill  the  holder  with  air,  and 
open  J:Tvery  slightly,  or  better,  allow  the  air  to  escape  through  a 
minute  aperture.  The  holder  will  now  slowly  descend,  and  by 
noting  the  time  the  index  passes  each  tenth  of  a  foot  mark,  a 
series  of  numbers  is  obtained  whose  first  differences  would  be  con- 
stant, if  the  apparatus  was  perfect.  By  varying  the  pressures,  the 
orifices  and  the  kind  of  gas  in  the  holder,  all  the  laws  of  the  flow 
of  gases  may  be  verified. 

57.     GAS-METERS. 

Apparatus.  Two  gas-meters,  one  wet  and  the  other  dry,  both 
graduated  so  as  to  read  to  thousandths  of  a  foot.  They  are  con- 
nected together  so  that  the  gas  will  pursue  the  following  course. 
It  leaves  the  pipe  through  a  \fr  valve,  passes  through  a  T  to  the 
wet  meter,  thence  through  a  second  T  to  the  dry  meter,  and  by 
a  stopcock  and  third  T  to  a  fishtail  burner.  A  short  piece  of  pipe 
is  screwed  into  the  open  end  of  each  T,  which  may  be  closed  by  a 
cap,  or  connected  with  a  gauge  formed  of  a  U-tube  by  a  piece  of 
rubber  tubing. 

Experiment.  Gas-meters  are  of  two  kinds,  wet  and  dry.  The 
former  consists  of  a  cylindrical  vessel  half  full  of  water,  in  which 
is  placed  a  rotary  drum  with  four  compartments.  As  these  are 
filled  in  turn,  the  drum  revolves,  and  the  amount  of  gas  con- 
sumed is  measured  by  the  number  of  revolutions.  The  dry  meter 
resembles  in  principle  a  blacksmith's  bellows  reversed  in  such  a 
way  that  air  being  forced  into  the  nozzle,  the  handle  moves  up  and 
down.  The  number  of  strokes  is  then  recorded  by  clockwork  and 
dials.  The  wet  meter  was  first  used,  but  is  now  superseded  in 
houses  by  the  dry  meter,  owing  to  the  error  introduced  by  any 
increase  or  diminution  in  the  amount  of  water  present.  The  former 
is,  however,  still  in  vogue  for  experiments,  as  by  it  small  amounts  of 
gas  may  be  measured  with  much  greater  accuracy.  To  determine 


112  GAS-METERS. 

the  amount  of  gas  which  has  passed  through  the  meter,  subtract  the 
reading  at  the  beginning  from  that  at  the  end  of  the  experiment, 
or  if  the  rate  of  flow  is  required,  take  readings  at  intervals  of  one 
minute,  as  in  Experiment  5.  Usually  in  the  best  meters,  one  rev- 
olution of  the  large  hand  equals  one  tenth  of  a  cubic  foot,  and  the 
dial  being  divided  into  a  hundred  parts  gives  thousandths.  In 
meters  intended  to  be  used  in  houses,  one  revolution  of  the  hands 
of  the  three  lower  dials  equals  100,000,  10,000  and  1000  cubic  feet, 
respectively,  and  a  fourth  dial  is  placed  above,  whose  hand 
makes  one  revolution  for  every  five  cubic  feet,  and  which  is  used 
in  testing  the  metre. 

The  common  method  of  testing  a  meter  is  to  bring  the  upper 
hand  to  the  zero,  connect  it  with  a4  gas-holder,  and  force  air  or  gas 
through  it  until  the  reading  is  the  same  as  before.  The  change  of 
reading  of  the  holder  should  now  be  just  five  feet,  and  the  differ- 
ence is  the  error  of  the  meter.  This  experiment  should  be  re- 
peated two  or  three  times.  If  the  meter  reads  to  thousandths  of 
a  foot  an  additional  test  is  needed  to  see  if  the  divisions  of  the 
large  dial  correspond  to  equal  quantities  of  gas.  For  this  purpose, 
allow  the  gas  to  flow  very  slowly  through  both  meters,  turn  it  off 
and  read  them,  dividing  the  thousandths  into  tenths  by  the  eye. 
Allow  a  few  thousandths  to  pass  and  read  again,  and  so  take  a 
series  of  readings,  until  two  complete  revolutions  have  been  made. 
Represent  the  results  by  a  residual  curve,  in  which  abscissas  repre- 
sent the  readings  of  the  large  hand  of  the  wet  meter,  and  ordi- 
nates  the  difference  between  the  two  enlarged,  moving  the  origin 
down  so  as  to  bring  the  points  on  the  paper.  Two  curves  are  thus 
obtained,  one  for  each  revolution,  which  should  be  coincident  ex- 
cept for  the  accidental  errors,  and  their  deviation  from  a  straight 
line  shows  the  inequality  of  the  thousandths,  as  given  by  the  dry 
meter. 

The  next  thing  to  be  determined  is  the  loss  of  pressure  due  to 
each  meter.  Evidently  a  certain  amount  of  power  is  necessary  to 
overcome  the  friction,  and  this  power  is  obtained  at  the  expense 
of  the  pressure  of  the  gas,  which  therefore  leaves  the  meter  with 
less  pressure  than  it  enters  it.  To  measure  this,  connect  the  first 
and  second  T  with  the  two  arms  of  the  gauge,  and  allow  the  gas 
to  pass  through  the  meter.  The  difference  in  level  of  the  two 


GAS-METERS.  113 

tubes  shows  the  loss  due  to  the  meter.  See  if  this  varies  with  dif- 
ferent pressures  and  with  different  positions  of  the  revolving  drum. 
By  using  the  second  and  third  T,  the  dry  meter  may  be  tested  in 
the  same  way. 

When  a  metre  is  placed  between  the  outlet  and  the  valve  by 
which  the  gas  is  turned  off,  an  error  is  introduced  whenever  the 
latter  is  opened  or  closed.  This  is  due  to  the  difference  of  pressure 
within  and  without  the  revolving  drum,  produced  by  gas  flowing 
in  or  out  without  in  some  cases  moving  the  hand.  To  show  this, 
close  both  the  valve  and  the  stopcock  near  the  third  T.  Open  the 
valve,  the  gas  will  rush  into  both  meters,  moving  the  hands  a 
small  amount.  Close  the  valve  and  open  the  stopcock,  when  the 
gas  will  rush  out  until  the  pressure  within  the  meter  equals  that 
of  the  outer  air.  Take  a  number  of  readings,  opening  them  thus 
alternately.  Each  meter  is  here  affected  by  the  error  caused  by 
the  other,  and  by  the  intermediate  pipe,  to  eliminate  which  the 
valve  and  stopcock  should  be  placed  close  to  the  meter  to  be 
tested.  The  error  may  then  be  determined  for  different  pres- 
sures and  different  positions  of  the  hand.  This  error  is  not  cu- 
mulative, and  seldom  exceeds  one  or  two  thousandths  of  a  foot. 

To  measure  the  amount  of  gas  consumed  by  any  burner  under 
different  pressures,  connect  it  with  one  of  the  meters,  and  attach 
the  gauge  to  the  third  T.  Turn  on  the  gas  and  take  a  five  minute 
observation ;  that  is,  take  six  consecutive  readings  of  the  meter  at 
intervals  of  one  minute,  also  the  pressure  as  given  by  the  gauge. 
Do  the  same  with  several  other  pressures,  and  see  if  the  flow  is 
proportional  to  the  square  root  of  the  pressure,  or  if  the  curve 
formed  by  the  readings  of  the  gauge  and  volumes  of  gas  burnt  is 
a  parabola.  A  similar  experiment  may  be  performed  with  an 
aperture  in  a  plate  of  platinum,  and  the  height  of  the  flame  meas- 
ured corresponding  to  different  pressures  and  rates  of  consumption. 
A  regulator  of  some  form,  such  as  will  be  described  under  the 
photometer,  should  be  introduced  to  prevent  accidental  variations 
in  the  pressure,  if  great  accuracy  is  expected  in  the  last  experi- 
ment. The  laws  of  the  efflux  of  gases  may  then  be  tested,  or  the 
uniform  division  of  the  meter,  by  allowing  the  gas  to  escape  very 
slowly,  and  seeing  if  the  volume  is  proportional  to  the  time. 

8 


114  BAROMETER. 

58.    BAROMETER. 

Apparatus.  Two  barometer  tubes,  one  already  filled  and 
placed  in  its  cistern,  some  pure  mercury,  and  a  stand  by  which  the 
height  of  the  mercury  column  may  be  measured.  This  stand  may 
be  made  in  a  variety  of  ways.  Thus  a  half  metre  steel  bar  divided 
into  millimetres  is  fastened  to  an  upright,  and  a  slider  attached  to 
it,  so  that  it  may  be  set  at  any  desired  height.  This  slider  carries, 
first  a  steel  point  about  40  centimetres  below,  to  determine  the 
height  of  the  mercury  in  the  cistern  ;  secondly,  a  vernier  or  a  brass 
plate  with  a  single  line  cut  on  it  and  resting  against  the  steel 
scale,  and  finally  two  index  plates  of  brass  between  which  the  tube 
is  placed.  The  slider  is  raised  or  lowered  until  a  thin  line  of  light 
is  just  visible  between  the  top  of  the  mercury  and  the  bottom  of 
the  index  plates,  and  the  reading  then  taken  by  the  vernier.  The 
tubes  are  held  in  position  either  by  rings  of  brass,  or  by  strips 
fastened  by  hinges.  A  steel  rod  three  or  four  decimetres  long  and 
a  tall  jar  of  water,  are  also  needed. 

Experiment.  First,  to  find  the  distance  from  the  steel  point  to 
the  index  plates.  This  may  be  done  by  the  cathetometer,  or  by 
the  second  steel  rod.  Place  the  jar  of  water  close  to  the  upright^ 
and  bring  the  points  of  both  rod  and  slider  just  in  contact  with 
the  surface  of  the  liquid.  Read  the  vernier,  lower  the  slider  until 
the  index  plates  are  just  on  a  level  with  the  top  of  the  steel  rod, 
and  read  again.  The  difference  added  to  the  length  of  the  rod 
equals  the  distance  from  the  index  plates  to  the  lower  point.  The 
length  of  the  steel  rod  is  found  by  bringing  the  index  plate  first  to 
its  upper  and  then  to  its  lower  end. 

Now  place  the  filled  barometer  in  its  proper  position  between 
the  index  plates,  move  the  slider  down  until  the  point  just  touches 
its  reflection  in  the  mercury,  and  read  the  vernier.  Raise  the 
slider,  until  placing  the  eye  on  a  level  with  the  index  plates  the 
light  is  just  cut  off  between  them  and  the  mercury.  The  differ- 
ence between  these  readings  is  the  height  through  which  the 
slider  has  been  raised,  and  this  added  to  the  distance  from  the 
plates  to  the  point  gives  the  height  of  the  mercury  column.  Now 
measure  the  height  of  the  standard  barometer  placed  with  the 
other  meteorological  instruments.  Reduce  it  to  millimetres  (1  me- 
tre =  39.37  inches),  and  the  difference  of  the  two  is  the  error  of 
the  barometer  first  measured.  It  is  probably  due  to  a  little  air  in 
the  top  of  the  tube. 


BAROMETER.  115 

Now  fill  the  empty  tube  in  the  following  manner.  If  the 
mercury  is  not  perfectly  pure,  it  must  be  cleaned  as  described  in 
Experiment  9.  Hold  the  tube  with  the  left  hand  in  an  inclined 
position,  the  closed  end  resting  on  the  table.  Pour  in  mercury 
slowly  to  within  a  few  inches  of  the  top.  To  prevent  spilling,  the 
stream  should  be  guided  by  the  forefinger  and  thumb  of  the  left 
hand  held  at  the  opening  of  the  tube.  Next  close  this  opening 
with  the  finger  and  raise  the  closed  end  so  that  the  bubble  of  air 
shall  move  slowly  along  the  tube.  Make  it  pass  from  end  to  end, 
until  all  the  small  adhering  air-bubbles  are  removed.  Then  fill  it 
full  of  mercury,  and  closing  the  end  again  invert  it,  and  immerse 
in  the  cistern,  removing  the  finger  under  the  surface  of  the  mer- 
cury. The  latter  will  now  descend  in  the  tube  until  its  height  is 
about  30  inches,  leaving  a  vacuum  at  the  top.  Its  pressure  at  the 
bottom  of  the  tube  is  then  just  equal  to  that  of  the  atmosphere, 
which  by  pressing  on  the  outside  mercury,  supports  the  inner  col- 
umn. The  vacuum  at  the  top  is  known  from  its  discoverer  as  the 
Torricellian  vacuum,  and  is  one  of  the  most  perfect  that  can  be  ob- 
tained artificially.  To  see  if  any  air  has  entered,  incline  the  tube, 
and  notice  if  the  mercury  rises  to  the  top,  remaining  at  the  same 
level  throughout,  and  if  when  made  to  oscillate  gently  it  strikes 
the  top  with  a  sharp  click ;  if  not,  air  has  entered,  and  the  experi- 
ment must  be  repeated.  Next,  put  this  tube  in  the  place  of  that 
previously  filled,  measure  its  height  and  determine  the  error. 

Allow  a  bubble  of  air  to  enter  one  of  the  barometer  tubes  (the 
one  in  which  the  error  is  the  greatest),  and  notice  that  it  increases 
in  volume  as  it  rises  until  it  reaches  the  top,  when  it  causes  a  con- 
siderable depression  of  the  mercury  column.  Repeat,  until  this 
has  fallen  eight  or  ten  inches.  Then  measure  the  height  with  care, 
and  suppose  it  to  be  an  observation  taken  at  the  top  of  a  moun- 
tain. Compute  the  height  on  this  supposition  by  the  method  given 
below.  The  temperature  of  the-  air  and  mercury  at  the  upper  sta- 
tion may  be  assumed  equal  to  that  of  the  thermometer  outside  the 
window,  and  at  the  lower  station  to  those  obtained  by  direct  ob- 
servation. This  work  may  well  be  supplemented  by  a  determina- 
tion of  the  altitude  of  a  real  mountain.  For  this  purpose  the 
pressure  of  the  air  must  be  measured  at  the  top  and  bottom,  either 
by  an  aneroid,  or  more  accurately  by  a  mercurial  mountain  barome- 


116          MEASUREMENT    OF    HEIGHTS    BY   THE    BAROMETER. 

ter.  When  going  on  such  an  expedition,  it  is  well  to  take  also  a 
hypsometer,  and  other  instruments,  so  as  to  determine  the  dew- 
point,  solar  radiation,  temperature  of  the  air,  etc.  These  will 
be  described  in  detail  under  Meteorological  Instruments.  On 
reaching  the  foot  of  the  mountain,  observations  should  be  taken, 
and  again  on  the  return,  and  the  mean  of  these  compared 
with  those  taken  at  the  top.  Or  better,  one  observer  with  a  ba- 
rometer is  left  below  to  take  readings  at  regular  intervals,  as  every 
quarter  of  an  hour,  during  the  whole  time  of  the  ascent.  These 
are  afterwards  compared  with  those  taken  at  the  same  time  at  the 
summit.  Of  course  the  lower  barometer  is  compared  carefully 
with  the  others  at  the  beginning  and  end  of  the  trip,  and  the 
errors  corrected.  If  only  one  barometer  is  at  hand,  and  time  al- 
lows, a  series  of  observations  should  be  taken  before  and  after  the 
ascent,  a  curve  constructed,  and  the  intermediate  readings  ob- 
tained by  interpolation.  Accuracy  is  to  be  expected  only  from  a 
long  series  of  observations  above  and  below,  by  which  accidental 
errors  are  eliminated ;  any  sudden  change  in  the  weather,  as  a 
thunder-storm,  is  especially  liable  to  affect  the  result. 

A  small  aneroid  which  may  be  easily  carried  in  the  pocket,  is 
often  very  serviceable  in  preliminary  surveys ;  by  using  it  in  con- 
nection with  a  pedometer,  an  approximate  profile  of  the  country 
may  be  constructed.  In  the  same  way  the  variations  in  the  grade' 
of  a  railway  may  be  determined.  The  delicacy  of  these  barom- 
eters is  such  that  they  will  show  the  difference  of  the  level  of  the 
different  parts  of  a  house,  or  even  the  rise  and  fall  of  a  vessel  at 
sea.  For  such  observations  the  height  is  obtained  with  sufficient 
accuracy  by  allowing  87  feet  for  every  tenth  of  an  inch  fall  of  the 
barometer. 

MEASUREMENT  OF  HEIGHTS  BY  THE  BAROMETER. 

On  ascending  from  the  surface  of  the  earth,  the  barometric  pres- 
sure continually  diminishes,  and  this  is  due  to  the  fact  that  being 
caused  by  the  weight  of  the  superincumbent  air,  the  greater  the 
height  the  less  the  load  to  be  borne.  The  law  of  diminution  is  easily 
deduced  by  the  calculus ;  call  p  the  pressure  in  inches  at  any  height 
J3~.  The  decrease  of  pressure,  or  — dp,  in  any  interval  dJT,  is  evi- 


MEASUREMENT    OF    HEIGHTS    BY    THE    BAROMETER.  117 

dently  due  to  a  column  of  air  of  this  height,  whose  weight  is  pro- 
portional to  p.  Hence  — dp  =  apdff,  dff=  —*-t  or  H  =  -  log  p 

+  O.  The  constant  a  equals  the  pressure  due  to  a  column  of  air 
of  height  unity  and  under  pressure  unity,  or  its  reciprocal  equals 
60,300.  The  elevation  E,  or  difference  in  height  of  two  points 
J^and  II'  is  therefore  H  —  Hr  =  60,300  (log/  —  log  p). 

In  order  to  obtain  the  true  height  by  the  formula,  it  is  necessary 
to  apply  several  corrections,  of  which  the  most  important  are  the 
following. 

I.  Capillarity.     The  effect  of  this  force  is  to  depress  the  mer- 
cury column  by  an  amount  dependent  on  the  diameter  of  the  tube. 
A  constant  quantity  should  therefore  be  added  to  each   reading. 
Unfortunately  this  result  is  modified  by  the  adhesion  of  the  liquid 
to  the  tube,  which  renders  this  correction  uncertain ;  sometimes, 
therefore,  the  height  of  the  meniscus  or  curved  portion  is  allowed 
for,  but  the  best  way  is  to  use  a  very  large  tube,  when  the  effect  of 
capillarity  becomes  inappreciable. 

II.  Temperature  of  the  Mercury.     The   standard  pressure  as- 
sumes the  temperature  to  be  0°  C. ;  at  higher  temperatures  the  mer- 
cury would  be  lighter,  and  the  pressure  less.    Let  p  be  the  observed 
height  at  temperature  T,  and  P  the  true  height  with  mercury  at 

*n 

zero,  then  p  =  P  (1  +  a  T),  or  P  —  +     . — m,  in  which  a  equals 

the  expansion  of  the  mercury  per  degree.  As  the  scale  expands 
also,  allowance  must  be  made  for  this,  which  gives  a  =  .00009, 
when  the  scale  is  of  brass.  The  temperature  T  is  given  by  the 
thermometer  attached  to  the  instrument,  and  this  correction 
should  always  be  applied  to  a  mercurial  barometer,  but  not  to  an 
aneroid,  when  the  height  is  wanted. 

III.  Temperature  of  the  Air.     In  the  above  discussion  the  air 
also  is  supposed  to  be  at  zero.     If  warmer  it  will  be  lighter,  and 
the  elevation  greater  than  that  here  assumed.     Call  t  and  if  the 
temperatures   above  and  below,  and  their  mean  t"  —  ^  (t  -\-  tf). 
The  true   elevation   E'  will  then  equal  E  (1   -f~  a  t")  in  which 
a  =  ^3-  the  coefficient  of  expansion  of  air.     This  is  the  most  im- 
portant correction  of   all,  and  should  always  be  applied,  or  large 
errors  will  be  introduced. 


118  BUNSEN   PUMP. 

IV.  Latitude.  Still  another  correction  may  be  applied  when 
great  accuracy  is  required,  owing  to  the  diminution  of  the  force 
of  gravity  as  we  approach  the  equator.  The  computed  elevation 
should  be  multiplied  by  (1  +  .0026  cos  2/),  in  which  I  is  the  lati- 
tude, since  the  force  of  gravity  varies  according  to  this  law. 

Introducing  these  corrections  into  the  formula  and  reducing,  it 
may  be  written  in  the  following  form, 

E  =  120(logp  —  log/)  (502  +  t  +  tf), 

which  may  be  applied  directly  to  observations  taken  with  an  ane- 
roid. For  a  mercurial  barometer,  p  and  p',  must  be  corrected,  first 
for  capillarity,  and  then  divided  by  (1  +  .00009  T)  and  (1  + 
.00009  Tr).  The  correction  for  latitude  is  always  small,  and  be- 
comes 0  at  45°. 

59.    BUNSEN  PUMP. 

Apparatus.  Fig.  46  represents  a  Bunsen  filter-pump,  such  as  is 
used  in  chemical  laboratories.  A  is"  a  valve  in  the  supply-pipe,  by 
which  the  water  is  admitted  to  the  bulb  IB.  From  this  it  with- 
draws a  portion  of  the  air,  which  passes  down  the  pipe  E  with  the 
water.  The  vessel  to  be  exhausted  (7,  is  connected  with  B  by 
the  long  pipe  (7Z?,  through  which  the  air  is  drawn.  Above  G  is 
placed  a  U  mercury-gauge,  and  below  it  a  wide  tube,  designed  to 
prevent  the  pressure  from  exceeding  a  certain  amount.  A  fine 
hole  is  made  near  the  bottom  of  this  tube,  and  it  dips  into  a  mer- 
cury-cistern D.  As  the  pressure  diminishes,  the  mercury  rises  in 
D  and  falls  in  the  outer  vessel  until  below  this  hole,  and  the  air 
rushes  in  and  increases  the  pressure ;  by  varying  the  height  of 
the  cistern  any  pressure  may  be  maintained.  This  device,  though 
excellent  in  theory,  often  gives  trouble  in  practice  from  the  jump- 
ing of  the  mercury,  unless  the  tube  is  large  and  the  hole  small. 
Instead,  therefore,  two  or  more  valves  may  be  used,  or  the  tube 
nearly  closed,  and  thus  the  air  admitted  so  slowly  as  to  keep  up 
the  required  pressure.  If  too  much  water  is  passed  through  B  it 
sometimes  overflows  into  O.  An  arm  and  stop  should  therefore 
be  attached  to  A,  so  that  it  cannot  be  opened  too  far.  The 
water  escaping  from  E  is  received  in  a  Florence  flask,  F,  which  is 
fitted  with  a  second  tube  G,  passing  nearly  to  the  bottom,  while 
E  opens  near  the  top.  To  measure  the  amount  of  water  expended, 
a  balance  and  weights  should  be  provided,  or  a  large  graduated 
vessel. 

Experiment.  Open  A  and  the  water  will  flow  through  .Z?,  and 
there  encountering  the  air,  will  carry  it  in  bubbles  through  E.  If 


BUNSEN    PUMP. 


119 


Fig.  46. 


now  C  is  closed,  the  air  will  gradually  be  carried  out  of  J9,  pro- 
ducing a  rarefaction,  and  the  air-bubbles 
in  E  will  be  found  to  occupy  less  and  less 
space  compared  with  the  water,  until  the 
limit  of  exhaustion  is  reached,  and  the 
tube  carries  off  nothing  but  water.  The 
diminution  in  pressure  thus  obtained 
should  nearly  equal  that  of  a  column  of 
water  of  height  BE,  or  if  this  is  made 
40  feet,  nearly  all  the  air  should  be  with- 
drawn. The  aqueous  vapor,  however,  al- 
ways remains,  and  for  other  reasons  the 
exhaustion  is  never  perfect;  it  neverthe- 
less forms  a  very  convenient  method  of 
producing  a  partial  vacuum. 

To  test  the  working  of  the  pump  and  its  efficiency,  the  follow- 
ing experiment  should  be  performed.  Pour  water  through  G  until 
F  is  filled  up  to  the  end  of  tho  tube  E.  Empty  it  by  blowing 
through  E,  collect  the  water  escaping  from  6?,  and  weigh  it.  The 
weight  in  grammes  gives  V^  the  volume  in  cubic  centimetres  of  the 
portion  of  F  included  between  the  ends  of  E  and  G.  Measure 
the  temperature  of  the  water  and  the  height  of  the  barometer.  Fill 
F  as  before,  take  C  out  of  the  mercury,  and  open  A  slightly.  A 
large  amount  of  air  and  a  small  amount  of  water  will  now  enter  the 
flask.  Water  will  flow  from  G  until  a  volume  of  air  equal  to  V 
has  entered  the  flask,  and  air  begins  to  bubble  up  through  G. 
Collect  the  water  that  has  escaped,  and  weigh  it.  Calling  its  vol- 
ume V,  the  amount  of  water  brought  from  B  is  evidently  V  — 
V,  while  in  the  same  time  "F  centimetres  of  air  have  been  brought 
down.  Record  also  the  time  required  to  empty  F.  Repeat  the 
experiment  several  times  with  a  larger  flow  of  water.  Try  also 
the  effect  of  a  flow  under  pressure  by  connecting  the  end  of  C  in 
the  mercury,  which  will  then  'rise  in  it  when  the  water  is  turned 
on,  and  may  be  kept  at  any  desired  height  by  raising  or  lowering 
D.  As  in  these  experiments  it  takes  some  time  for  the  mercury 
to  attain  its  normal  level,  it  is  well  to  connect  a  third  tube  with  the. 
flask,  which  may  then  be  filled  without  disconnecting  it  from  JS. 
It  will  be  seen  that  the  best  results  are  attained  when  the  smallest 


120  AIR-METER. 

amount  of  water  is  used,  but  as  the  exhaustion  then  takes  place 
very  slowly,  it  is  often  best  to  begin  with  a  large  flow,  and  dimin- 
ish it  as  the  air  is  withdrawn.  The  maximum  amount  of  air  that 
might  be  drawn  out  by  the  apparatus  may  be  determined  analyt- 
ically, and  dividing  the  observed  amount  by  this,  gives  the  effi- 
ciency. 

This  same  apparatus  may  also  be  employed  with  advantage,  to 
test  aneroid  barometers.  G  is  attached  to  an  air-tight  chamber, 
formed  of  a  tubulated  receiver  placed  on  an  air-pump  plate.  When 
the  water  is  turned  on,  the  air  is  gradually  withdrawn,  and  the 
barometer  falls.  The  reading  is  compared  with  the  true  pressure 
found  by  subtracting  the  reading  of  the  U  gauge  from  the  height 
of  the  standard  barometer.  Different  results  will  be  attained  ac- 
cording as  the  barometer  is  placed  vertically  or  horizontally,  or 
if  the  friction  is  reduced  by  gently  tapping  on  the  instrument. 
To  render  the  test  more  complete  this  experiment  should  be  tried 
at  different  temperatures,  which  is  best  effected  by  a  water  jacket, 
which  may  be  filled  either  with  hot  or  cold  water. 

60.    AIR-METER. 

Apparatus.  An  organ  bellows,  such  as  is  described  in  the  next 
experiment,  and  an  air-meter,  of  which  a  very  convenient  form  is 
that  manufactured  by  Casella.  It  consists  of  a  very  light  fan- 
wheel,  like  a  wind-mill,  with  a  counter  to  record  the  number  of 
revolutions.  The  vanes  are  set  at  such  an  angle  that  the  divisions 
of  the  dial  shall  represent  the  number  of  feet  traversed  by  the 
air. 

Experiment.  Work  the  bellows  and  allow  the  air  to  escape 
through  an  orifice,  so  as  to  produce  a  constant  current  of  air. 
Measure  its  velocity  at  intervals  of  ten  inches  from  the  orifice, 
until  it  becomes  imperceptible.  Measure  also  the  velocity  on  each 
side  of  the  central  line.  A  spring  catch  serves  to  throw  the  gear- 
ing in  or  out  of  connection  with  the  fan-wheel.  To  make  an 
observation,  therefore,  the  meter  should  be  placed  in  the  current, 
and  when  it  has  attained  a  uniform  velocity  thrown  in  gear  for 
'exactly  one  minute.  As  the  hands  move  only  during  this  time, 
the  difference  of  readings  taken  before  and  after,  give  the  distance 
traversed,  or  dividing  by  60,  the  velocity  of  the  air  per  second. 


AIR    METER.  121 

A  table  of  corrections  accompanies  the  instrument,  showing  how 
much  should  be  added  or  subtracted  from  the  observed  readings 
to  get  the  true  distance. 

Interesting  results  may  be  attained  with  this  instrument  on  the 
velocity  of  the  wind,  especially  during  gales,  the  air  currents  in 
buildings  from  registers,  ventilators  or  doors  slightly  open.  This 
forms  one  of  the  most  efficient  means  of  studying  the  ventilation 
of  large  halls  and  churches. 


SOUND. 


61.      SlRENE. 

Apparatus.  An  organ  bellows  capable  of  giving  a  perfectly 
constant  current  of  air  under  various  pressures.  One  of  the  best 
forms  is  that  made  by  Cavaille  Coll  (sold  by  Konig)  with  regu- 
lator attached.  If  preferred,  a  large  gas-regulator  may  be  at- 
tached to  any  bellows.  A  set  of  organ-pipes  well  tuned,  giving 
the  notes  of  the  scale  from  C3  to  C"4,  two  or  three  tuning  forks, 
one  giving  the  French  normal  pitch,  etc.  and  a  sirene.  The  lat- 
ter need  not  be  of  large  size,  as  good  results  may  be  obtained 
with  a  single  moving  disk  with  one  circle  of  holes. 

Experiment.  Place  the  organ-pipe,  <73,  in  its  hole  on  the  bel- 
lows, and  connect  the  sirene  so  that  the  air  shall  pass  through  it. 
Work  the  bellows,  and  the  perforated  disk  will  begin  to  revolve,  at 
first  slowly,  giving  a  rustling  or  humming  sound,  and  then  faster, 
producing  a  note  of  low  musical  pitch.  As  the  speed  increases 
the  pitch  rises,  until  it  is  about  that  of  the  pipe.  Sound  the  lat- 
ter, and  increase  or  diminish  the  pressure  of  the  air,  so  that  they 
shall  be  precisely  together.  A  slight  deviation  produces  beats, 
that  is,  an  alternate  increase  and  diminution  in  the  intensity  of  the 
sound  for  every  vibration  gained  or  lost  by  the  sirene.  By  a  little 
practice  these  beats  may  be  made  to  take  place  very  slowly,  or  not 
at  all.  The  wheelwork  of  the  sirene  may  be  thrown  in  or  out  of 
gear  with  the  revolving  shaft,  so  that  the  hands  may  or  may  not 
register  the  number  of  turns  of  the  perforated  disk.  Throw  it 
out  of  gear,  and  read  the  position  of  the  hands.  Bring  the  two 
sounds  in  unison,  and  keep  them  together  for  a  minute,  during 
which  time  the  shaft  is  thrown  in  gear,  and  the  hands  are  moving. 
The  difference  of  the  readings  before  and  after,  gives  the  number 
of  turns,  and  this  multiplied  by  the  number  of  holes  in  the  per- 
forated disk,  give  the  number  of  complete  vibrations.  Dividing 

122 


KUNDT'S  EXPERIMENT.  123 

by  60  gives  the  number  per  second.  Repeat  with  the  other  pipes, 
and  see  if  this  ratio  is  that  given  by  theory.  Do  the  same  with 
the  tuning  forks.  This  is  more  difficult  as  they  cannot  be  sounded 
continuously.  The  best  method  of  sounding  a  tuning  fork  is  by 
means  of  a  violin  bow.  The  latter  should  be  held  near  the  end 
of  the  fork,  nearly  parallel  to  the  two  prongs,  but  touching  only 
one,  and  drawn  with  considerable  pressure,  and  not  too  rapidly 
To  prevent  slipping  it  should  be  well  rubbed  with  resin. 

62.     KTJNDT'S  EXPERIMENT. 

Apparatus.  Several  glass  tubes  two  or  three  inches  in  diame- 
ter, and  six  feet  long,  one  open  at  both  ends,  the  others  closed  and 
filled  with  different  gases,  and  also  containing  a  little  lycopodium 
powder.  A  number  of  rods  of  brass,  steel,  glass  and  wood,  and  a 
clamp  by  which  they  may  be  held  at  the  centre.  Three  of  the 
rods  should  be  of  the  same  material,  but  one  of  double  diameter, 
the  second  half  the  length,  of  the  third.  Cloths  which  may  be  wet 
or  covered  with  resin  should  be  provided  to  set  them  in  vibration, 
also  some  lycopodium  powder. 

Experiment.  Place  a  little  lycopodium  powder  in  the  open 
tube,  hold  it  horizontally  by  the  middle,  and  rub  it  lengthwise 
with  a  wet  cloth.  A  clear  musical  note  of  high  pitch  is  at  once 
produced,  and  the  powder  arranges  itself  in  about  fifteen  to 
twenty  groups  at  regular  intervals  along  the  tube.  The  reason  is, 
that  the  air  in  the  interior  of  the  tube  vibrates  with  the  same 
rapidity  as  the  glass,  but  as  the  velocity  of  sound  in  it  is  much 
less,  the  wave-length  is  less  in  the  same  proportion.  Hence  divid- 
ing the  length  of  the  tube  by  the  distance  apart  of  the  lycopo- 
dium groups  gives  the  relative  velocity  of  sound  in  glass  and  air, 
or  multiplying  this  number  by  33&  gives  the  velocity  of  sound  in 
glass  in  metres. 

If  the  tube  is  filled  with  any  other  gas  than  air  the  interval  will 
be  proportional  to  the  velocity.  Thus  knowing  the  velocity  in 
glass,  the  velocity  in  the  gas  may  be  obtained.  Make  this  meas- 
urement with  the  other  tubes,  and  see  if  the  law  holds  that  the 
velocity  is  inversely  proportional  to  the  square  root  of  the  density. 

This  same  method  may  be  applied  to  the  accurate  determination 
of  the  velocity  of  sound  in  solids.  One  of  the  rods  is  clamped  at 
the  centre,  and  the  end  inserted  in  the  open  glass  tube.  The  air 


124 


MELDE  S    EXPERIMENT. 


in  the  latter  is  confined  by  a  cork  at  one  end,  and  a  disk  some- 
what smaller  than  the  tube  is  attached  to  the  rod.  The  latter  is 
now  set  in  vibration  by  a  cloth'  moistened  with  water,  for  glass,  or 
covered  with  resin,  for  wood  or  metal.  The  vibrations  of  the  rod 
are  transmitted  to  the  air,  and  the  heaps  of  sand  formed.  In 
general  these  will  not  be  clearly  defined,  because  the  whole  length 
of  the  air  space  is  not  an  exact  multiple  of  half  a  wave-length. 
The  rod  should  therefore  be  moved  in  or  out  until  the  heaps  are 
distinctly  marked.  The  velocity  of  sound  in  the  rod  is  then  ob- 
tained by  the  following  .calculation.  If  L  is  the  length  of  rod, 
I  the  distance  between  the  heaps  of  lycopodium,  and  V  = 
333  (1  +  .0037  £)2,  the  velocity  of  sound  in  air  at  any  tempera- 
ture t,  then  the  velocity  in  the  rod  =  — j- .  The  temperature  t 
may  be  taken  as  equal  to  that  of  the  room,  and  measured  with  a 
Centigrade  thermometer. 

63.    MELDE'S  EXPERIMENT. 

Apparatus.  A  tuning-fork  projecting  horizontally  from  a  verti- 
cal wall,  and  tuned  to  give  a  low  note,  as  G^.  Four  weights  in 
the  ratio  1,  £,  £,  fy,  made  of  brass  rods  cut  to  these  lengths  respec- 
tively, some  fine  silk  thread,  and  a  millimetre  scale.  A  piece  of 
brass  with  a  hole  in  it  should  be  fastened  to  the  end  of  one  prong 
of  the  fork,  and  a  fine  wire  hook  attached  to  the  silk  to  support 
the  weights.  A  violin  bow  is  also  needed  to  excite  the  fork,  or 
bette,  ran  electro-magnetic  attachment,  by  which  the  vibrations 
may  be  maintained  continuously. 

Experiment.  By  this  apparatus  the  various  laws  for  the  vibra- 
tions of  cords  may  be  proved.  1st.  The  time  of  vibration  is 
proportional  to  the  length.  Place  the  fork  so  that  its  two  prongs 
shall  lie  in  the  same  vertical  plane,  and  suspend  the  largest  weight 
from  it  by  the  silk  thread.  Sound  the  fork,  as  described  in  Ex- 
periment 61,  and  vary  the  length  of  the  thread  until  its  time  of 
vibration  corresponds  with  that  of  the  fork.  When  this  is  the 
case  it  will  form  a  loop  or  spindle,  fixed  at  the  ends  and  swelling 
out  at  the  centre  through  several  inches.  As  this  occurs  only 
when  the  cord  is  very  nearly  the  right  length  it  may  be  tuned 
quite  accurately  by  the  eye  alone.  Make  three  or  four  observa- 
tions in  this  way,  measuring  the  length  in  each  case  with  the  mil- 


ACOUSTIC    CURVES.  125 

limetre  scale.  Next  turn  the  fork  90°,  so  that  the  prongs  shall  lie 
in  the  same  horizontal  plane.  The  cord  will  now  make  as  many 
vibrations  as  the  fork,  while  in  the  former  case  it  made  but  half  as 
many.  This  is  evident  if  the  relative  positions  of  the  prong  and 
cord  are  compared.  When  the  prong  is  in  its  highest  position  the 
cord  is  straight  or  central.  As  the  prong  descends  it  moves  to 
the  right,  and  as  it  ascends  again  becomes  central.  At  the  next 
descent  of  the  prong  it  moves  to  the  left,  becoming  central  a  second 
time,  when  the  prong  has  reached  the  top.  It  thus  makes  only  one 
complete  vibration,  while  the  fork  makes  ,two.  Accordingly  when 
the  fork  is  turned  around  90°,£he  cord  will  be  set  vibrating  ex- 
actly twice  as  fast  as  before.  On  trying  the  experiment  it  will  be 
found  that  the  new  length  of  cord  required  will  be  just  one  half 
that  in  the  first  case.  That  is,  a  double  length  requires  a  double 
time. 

2d.  The  time  of  vibration  is  inversely  proportional  to  the 
square  root  of  the  tension.  Applying  the  four  weights  in  succes- 
sion, the  corresponding  lengths  are  proportional  to  1, 2,  3  and  4,  or 
since  for  equal  tensions  the  times  are  proportional  to  the  lengths, 
the  law  is  proved. 

3d.  The  time  is  proportional  to  the  diameter  of  the  cord.  A 
second  cord  of  precisely  twice  the  diameter  of  the  first,  may  be 
made  by  twisting  four  strands  of  the  former.  It  will  be  found 
that  the  length  must  be  reduced  one  half  to  obtain  the  same 
effect  as  before. 

All  these  laws  may  also  be  proved  by  preparing  a  string  of  such 
a  length  that  it  will  vibrate  as  a  whole,  when  the  larger  weight  is 
applied,  then  attaching  the  other  weights  it  divides  into  2,  3  or  4 
loops,  separated  by  fixed  points  or  nodes,  and  corresponding  to  the 
harmonics  of  the  cord.  A  second  string  of  double  thickness 
serves  to  prove  the  3d  law.  A  simple  proof  of  the  first  law  may 
also  be  obtained  by  a  second  fork  an  octave  higher  than  the  other. 

64.    ACOUSTIC  CURVES. 

Apparatus.  In  Fig.  47,  A  is  a  large  tuning-fork  capable  of  giv- 
ing out  at  least  one  harmonic  besides  its  fundamental  note,  and 
carrying  on  the  end  of  one  of  its  prongs  a  piece  of  sheet  brass  cut 
to  a  point.  B  is  a  little  carriage  on  which  a  piece  of  smoked 


JFD 


126  ACOUSTIC    CURVES. 

glass  may  be  laid  and  drawn  under  the  brass  point  or  style  by  a 
cord  passing  over  the  pulley  C.  Two  weights,  D  and  E,  are  at- 
tached below,  the  upper  one,  D,  being  just  equal  to  the  friction  of 
the  carriage.  Some  pieces  of  glass  about  three  inches  by  four,  are 
needed,  and  a  gas-burner,  by  which  they  may  be  covered  with 
lampblack.  By  using  the  size  of  glass  employed  in  the  lantern 
for  projections,  the  curves  may  be  thrown  on  the  screen  on  a 
greatly  enlarged  scale. 

Experiment.  Cover  one  of  the  plates  of  glass  with  a  layer  of 
lampblack  by  holding  it  by  one  corner  over  the  gas-flame,  and 

moving  it  about  so  that  the 
coating  shall  be  uniform,  and 
very  thin.  Instead  of  lamp- 
black, collodion  may  be  used, 
pouring  it  on  in  the  usual 
way,  as  when  taking  a  photo- 
graph. Care  must  be  taken 
Fis-47-  to  select  such  collodion  as 

will  give  an  opaque  and  very  tender  film,  when  results  of  extreme 
beauty  and  delicacy  will  be  obtained.  Lay  the  glass  down  on  the 
carriage,  and  raise  it  so  that  when  passed  under  the  style,  "the 
latter  will  just  touch  its  surface.  This  may  be  accomplished  by 
wedges  or  levelling  screws  under  the  glass.  Draw  JB  back  a  short 
distance  beyond  the  style,  and  release  it,  when  it  will  begin  to 
move  under  the  action  of  the  two  weights  D  and  E.  The  length 
of  the  cord  should  be  such  that  when  the  wagon  reaches  the  style, 
E  will  touch  the  floor  so  that  the  carriage  will  move  with  a  uni- 
form motion  by  its  inertia,  the  friction  being  just  compensated  by 
D.  The  style  will  accordingly  draw  a  fine  unbroken  straight  line 
over  the  glass.  Now  sound  the  fork  by  the  violin  bow  (see  Ex- 
periment 61),  and  again  pass  the  carriage  under,  when  the  line, 
instead  of  being  straight,  will  be  marked  by  sinuosities,  one  cor- 
responding to  each  vibration  of  the  fork. 

Next  sound  the  harmonic,  by  drawing  the  bow  somewhat  more 
rapidly,  and  with  less  pressure  than  before,  at  a  point  about  two- 
thirds  of  the  distance  from  the  end  of  the  prong  to  the  handle. 
The  sound  sometimes  comes  out  more  readily  by  lightly  touching 
the  intermediate  one-third  point  or  node  with  the  finger.  A  high, 
clear  note  is  thus  produced,  and  on  drawing  the  carriage  back  the 


ACOUSTIC    CURVES.  127 

same  distance  as  before,  and  letting  it  again  pass  under,  another 
curve  is  obtained,  with  indentations  much  nearer  together,  owing 
to  the  greater  rapidity  of  the  undulations.  Of  course  the  plate  is 
moved  sideways  a  short  distance  each  time,  to  prevent  the  curves 
from  overlapping.  Produce  the  fundamental  note,  and  while  it  is 
sounding  draw  the  bow  so  as  to  give  the  harmonic,  and  immedi- 
ately let  go  the  carriage.  A  curve  is  thus  obtained,  resulting  from 
these  two  systems  of  vibrations,  and  consisting  of  small  sinuosities 
superimposed  on  larger  ones.  Determine  their  ratio  by  seeing 
how  many  of  the  former  correspond  to  an  exact  number  of  the 
latter.  Write  on  the  lampblack  your  name  and  the  date,  and  if 
all  the  curves  are  good,  varnish  the  plates  to  render  them  perma- 
nent. For  this  purpose  expose  the  blackened  surface  to  the  vapor 
of  boiling  alcohol  to  remove  the  grease,  then  holding  it  by  one 
corner  pour  amber  varnish  over  it  precisely  as  when  varnishing  a 
photographic  negative. 

To  compare  the  lines  with  theory,  place  the  glass  in  a  magic 
lantern,  and  project  an  image  of  it  on  the  screen.  If  the  sun  is 
used  as  a  source  of  light,  it  is  scarcely  necessary  to  darken  the 
room.  Place  a  sheet  of  paper  so  that  three  or  four  undulations  of 
the  curve  of  the  fundamental  note  shall  fall  on  it.  Trace. them 
carefully  with  a  pencil  and  an  enlarged  reproduction  of  the  origi- 
nal is  obtained.  Draw  lines  tangent  to  the  waves  above  and 
below,  and  bisect  the  space  between  them  by  a  line.  It  will 
intersect  the  curve  in  points  at  regular  intervals  #,  any  one  of 
which  may  be  taken  as  the  origin  of  coordinates.  If  a  is  the 
height  of  the  wave,  or  one  half  the  distance  between  the  two 

-~/y» 

tangent  lines,  the  theoretical  equation  will  be  y  =  a  sin  -r- .  Con- 
struct points  of  this  curve  by  dividing  the  space  between  two 
consecutive  intersections  of  the  curve  into  six  equal  parts,  and  lay 
off  vertical  distances  equal  to  a  multiplied  by  sin  15°,  30°,  45°, 
etc.,  to  180°.  These  sines  have  the  following  values: —  sin  15°  = 
.259,  sin  30°  =  .500,  sin  45°  =  .707,  sin  60°  =  .866,  sin  75°  =  .966. 
Draw  a  smooth  curve  through  the  points  thus  obtained,  and  com- 
pare it  with  that  given  by  the  forks.  To  test  the  combination  of 
the  two  systems  of  vibrations  is  more  difficult,  but  it  may  be  done 


128  LISSAJOUS'  EXPERIMENT. 

by  taking  their   equations  separately,  yf  —  a  sin  -T-,    and   y"  = 
ar  sin  -^7,  and  adding  them  so  that  y  =•  y'  +  y"  =  a  sin   V     + 

a'  sin  ~. 

An  immense  variety  of  curves  may  be  obtained  by  mounting 
the  plate  on  a  second  tuning-fork,  which  is  also  set  vibrating. 
Different  curves  are  thus  obtained,  according  as  the  motion  of  the 
style  is  parallel  or  perpendicular  to  the  vibrations  of  the  plate, 
also  with  every  change  in  the  interval  between  the  two  forks. 
With  this  arrangement  it  is  much  better  to  maintain  the  vibra- 
tions of  one  or  both  forks  continuously  by  electricity.  Better 
effects  are  also  obtained  in  this  way  in  Melde's  and  Lissajous'  ex- 
periments. 

Instead  of  projecting  the  curve  on  the  screen  it  may  be  measured 
by  the  Dividing  Engine,  Experiment  21,  or  enlarged  by  a  micro- 
scope and  drawn  by  a  camera  lucida,  The  length  of  the  waves 
gives  a  very  delicate  test  of  the  uniformity  of  the  motion  of  the 
car,  a  difference  of  a  ten  thousandth  of  a  second  being  easily 
perceived. 

65.    LISSAJOUS'  EXPERIMENT. 

Apparatus.  Mirrors  are  attached  to  the  ends  of  the  prongs  of 
two  tuning-forks,  and  the  image  of  a  spot  of  light  reflected  in 
them  is  viewed  in  a  telescope.  The  planes  of  the  tuning-forks 
must  be  perpendicular,  that  is,  one  must  vibrate  in  a  vertical,  the 
other  in  a  horizontal  plane.  It  is  best  to  have  a  series  of  forks 
with  sliding  weights,  so  that  all  the  intervals  in  the  octave 
may  be  obtained.  A  good  spot  of  light  is  produced  by  a  gas  flame 
shining  through  a  small  aperture  in  a  metallic  plate,  or  a  mirror 
may  be  used  to  reflect  the  light  of  the  sky. 

Experiment.  On  looking  through  the  telescope  a  minute  spot 
of  light  should  be  visible.  When  one  of  the  tuning-forks  is 
sounded  the  mirror  is  moved  from  side  to  side,  carrying  the  image 
of  the  spot  with  it  so  rapidly  as  to  make  it  appear  like  a  horizon- 
tal line  of  light.  In  the  same  way  the  motion  of  the  other  fork 
produces  a  vertical  line.  When  both  sound,  a  curve  is  formed, 
which  remains  unchanged  if  the  concord  is  exact,  but  continually 
alters  if  the  forks  are  not  a  perfect  tune.  Bring  the  forks  in  uni- 


LISSAJOUS'    EXPERIMENT.  129 

son  by  placing  the  weights  on  the  corresponding  points  of  each. 
They  are  best  sounded  by  a  bass-viol  bow,  drawing  it  slowly  and 
with  pressure  over  the  end  of  one  prong  nearly  parallel  to,  but  not 
touching,  the  other.  As  the  bows  soon  wear  out  by  the  horsehair 
giving  way,  a  convenient  and  cheap  substitute  is  made  by  covering 
a  strip  of  wood  of  proper  shape  with  leather,  which  when  rubbed 
with  resin,  answers  very  well. 

On  sounding  both  forks,  having  brought  them  in  unison  as 
above,  the  point  of  light  is  in  general  converted  into  an  ellipse 
which,  as  it  is  impossible  to  tune  them  exactly  by  the  ear,  grad- 
ually changes  into  a  straight  line,  then  into  an  ellipse,  a  circle,  an 
ellipse  turned  the  other  way,  a  straight  line  and  so  on.  Raise  the 
pitch  of  one  of  the  forks  slightly,  by  moving  the  weight  towards 
the  handle,  and  if  the  changes  take  place  more  slowly  the  unison 
is  more  perfect.  By  trial,  first  moving  the  weights  one  way  and 
then  the  other,  they  may  be  brought  in  tune  with  any  desired 
degree  of  exactness,  and  far  nearer  than  is  possible  by  the  ear 
alone,  as  the  complete  change  of  the  curve  from  one  line  to  the 
other  denotes  that  one  fork  has  advanced  only  a  single  vibration. 

Next  make  one  fork  the  octave  of  the  other,  and  a  curve  is  ob- 
tained, changing  from  the  parabola  to  the  lemnescata,  or  figure  8. 
A  simple  rule  serves  to  determine  the  interval  in  all  cases  from  the 
curve.  Count  the  number  of  points  where  the  latter  touches  the 
sides  of  the  rectangle  bounding  it,  also  the  number  of  points 
where  it  touches  its  top  or  bottom ;  the  ratio  of  these  two  is  the 
interval  between  the  forks.  When  the  curve 
terminates  in  either  corner  this  point  must  be 
counted  as  one  half  on  the  horizontal,  and  half 
on  the  vertical,  bounding  line.  Thus  in  Figure 
48,  both  A  and  B  correspond  to  the  ratio  of 
2  :  3,  or  the  interval  of  the  fifth. 

The  more  perfect  the  concord  the  more  slowly  will  the  curves 
alter  their  form,  and  the  simpler  the  ratio  of  the  number  of  vibra- 
tions the  simpler  the  curve.  When  the  forks  are  not  quite  in 
unison,  beats  will  be  heard,  and  the  curve  will  then  be  seen  to 
alter  its  form  so  as  to  keep  time  with  them.  Next  try  some  other 
ratios,  as  f ,  f,  f ,  &  4»  i  >  also  some  more  complex  curves,  as  f ,  f , 
and  -. 


130  CHLADNI'S    EXPERIMENT. 

66.     CHLADNI'S  EXPERIMENT. 

Apparatus.  A  number  of  brass  plates  attached  to  a  stand,  a 
violin  bow  and  some  sand.  A  good  series  of  plates  consists  of 
three  circles,  whose  diameters  are  as  2  :  2  :  1,  and  their  thickness 
as  1  :  2  :  1.  Also  three  square  plates,  similarly  proportioned. 

^Experiment.  The  plates  are  sounded  by  touching  them  at  cer- 
tain points  and  drawing  the  bow  across  their  edges,  holding  it 
nearly  vertical,  and  moving  it  slowly  and  with  considerable  pres- 
sure. A  sound  is  thus  produced,  and  certain  lines  are  formed  on 
the  plate  called  nodal  lines,  which  remain  at  rest,  the  other  parts 
vibrating.  If  sand  is  sprinkled  uniformly  over  the  plate,  that  on 
the  nodal  lines  will  remain  there,  the  rest  being  thrown  up  and 
down,  so  that  finally  it  will  all  collect  on  these  lines.  The  higher 
the  note  the  more  complex  the  nodal  lines,  and  the  nearer  they 
are  together. 

Taking  first  the  largest  and  thinnest  circular  plate,  touch  it  at 
any  point  of  the  circumference,  and  bow  it  at  a  point  about  45° 
distant.  The  sand  will  collect  on  two  lines  at  light  angles.  Next 
bow  it  at  a  point  90°  distant,  and  it  will  divide  into  six  parts.  By 
touching  the  plate  at  two  points  distant  45°  with  the  thumb  and 
middle  finger  of  the  left  hand,  and  bowing  the  point  midway  be- 
tween them,  a  division  into  eight  equal  parts  is  obtained.  In  the 
same  way  10,  12,  or  any  even  number  of  parts  are  formed,  until 
the  divisions  become  so  small  that  they  cannot  be  sounded. 

Next  try  the  first  square  plate.  The  lowest  sound  this  will 
give  is  obtained  by  touching  the  centre  of  one  side,  and  bowing 
the  corner.  The  next  note,  a  fifth  above,  is  produced,  when  the 
corner  is  held  and  the  centre  bowed.  By  altering  the  position  of 
the  fingers  and  bow,  a  great  variety  of  figures  may  be  obtained, 
which  may  be  still  further  extended  by  changing  the  points  of 
support,  or  the  form  of  the  plate.  Moreover,  among  plates  of  the 
same  shape  some  seem  to  give  out  certain  curves  more  easily 
than  others,  owing  probably  to  peculiarities  in  their  internal  struc- 
ture. A  square  plate  generally  gives  readily,  besides  the  curves 
described  above,  one  formed  of  two  diagonal  lines,  and  four  half 
ovals  on  its  edges.  The  pitch  is  three  octaves  above  that  of  the 
diagonal  lines  alone.  Another  curve  of  extreme  beauty  consists 
of  a  circle  with  eight  radial  lines,  and  the  intermediate  spaces 


CHLADNIS    EXPERIMENT.  131 

marked  by  eight  half  ovals.    It  will  be  noticed  that  the  points 
touched  are  necessarily  at  rest,  and  hence  lie  on  the  nodal  lines. 

By  using  the  three  plates  of  the  same  form,  some  of  the  laws  of 
the  vibrations  of  plates  may  be  proved.  1st.  The  number  of 
vibrations  is  proportional  to  the  thickness.  Form  the  same  curve 
on  the  two  plates,  of  which  one  is  double  the  thickness  of  the 
other,  and  it  will  be  noticed  that  the  pitch  is  always  an  octave 
higher.  2d.  In  similar  plates  of  equal  thickness,  the  number  of 
vibrations  is  inversely  as  the  square  of  the  homologous  parts. 
Hence  the  small  plate  gives  a  note  two  octaves  above  the  large 
one  of  the  same  thickness. 


LIGHT. 


67.    PHOTOMETER  FOR  ABSORPTION. 

Apparatus.  In  Fig.  59,  A  is  the  source  of  light,  which  may  be 
a  candle  in  a  spring  candle-stick,  or  a  small  gas  jet.  B  and  C  are 
two  mirrors  set  at  such  an  angle  that  they  will  form  images  of  A, 
just  50  inches  apart,  and  making  ABC  a  right-angled  triangle.  D 
is  a  Bunsen  photometer  disk,  made  by  placing  a  circular  piece  of 
thick  paper  in  a  lathe,  and  painting  all  but  the  centre  with  the 
best  melted  sperm-candle  wax.  It  then  possesses  the  property 
when  placed  between  two  lights,  of  changing  its  appearance  ac- 
cording as  one  or  the  other  is  the  brighter.  It  is  mounted  on  a 
slide  and  carries  an  index  which  moves  over  a  graduated  scale. 
F  is  a  screen  so  placed  as  to  protect  D  and  the  eyes  of  the  ob- 
server from  the  direct  light  of  A,  while  it  leaves  the  scale  illumin- 
ated so  that  it  can  be  easily  read.  A  stand  with  a  graduated 
circle  is  also  provided,  on  which  one  or  more  plates  of  glass  may 
be  set  and  inclined  at  any  angle  to  the  ray  of  light  AJB.  Some 
observers  prefer  a  disk  with  only  the  central  part  covered  with 
wax,  and  instead  of  a  circular  spot  use  some  other  form.  The 
great  difficulty  in  these  cases  is  to  distribute  the  wax  uniformly, 
and  prevent  its  accumulating  at  the  edges.  Still  another  method 
is  to  punch  figures  in  a  sheet  of  thick  paper,  and  cover  both  sides 
of  it  with  tissue  paper,  taking  care  that  no  wrinkles  remain.  The 
whole  apparatus  must  be  used  in  a  darkened  room. 

Experiment.     The  disk  D  possesses  the   curious  property  of 

appearing  bright  in  the  centre  when 
a  strong  light  is  in  front  of  it,  but 
dark  when  the  brighter  light  is  be- 
hind. When  placed  between  two 
lights  there  is  therefore  a  certain 
position  where  the  spot  will  disppear, 
in  which  case  it  is  so  much  nearer 
the  fainter  light  that  the  illumination 
on  its  two  sides  are  equal.  Their  rel- 
ative brightness  may  then  be  readily 
132 


PHOTOMETER    FOR    ABSORPTION.  133 

determined  from  the  law  that  the  intensity  is  inversely  as  the  square 
of  the  distance.  Now  the  two  images  of  A  act  like  two  precisely 
similar  lights,  any  change  in  one  affecting  the  other  equally.  By 
moving  D  the  centre  spot  may  be  made  either  light  or  dark,  and 
there  will  be  a  certain  intermediate  position  in  which  it  will  dis- 
appear. The  exact  point  of  disappearance  can  be  determined  only 
by  long  practice,  noticing  that  it  varies  with  the  position  of  the 
eye,  and  with  the  two  sides  of  the  disk.  Find  this  point  as  nearly 
as  possible,  read  the  index,  move  D  a  short  distance,  set  again 
and  take  the  mean  of  several  such  observations.  Compute 
the  probable  error  in  inches,  and  the  result  multiplied  by  4 
gives  the  error  in  percentage.  Let  x  be  the  mean  observed 
reading,  or  AB  +  BD.  Then  the  distance  of  the  other  image  of 
the  light  equals  DC  +  OA^  or  50  —  x.  Calling  B  and  O  their 
intensities  at  a  distance  unity,  their  intensities  at  the  distance  of 

7?  C1 

the  disk  will  be  —3   and    ,*„      — c-2 ;  or  since  these  quantities  are 

B         /      x      \2 
equal,  their  relative  intensities  I  =  -^  =  (  ^-— —  i  .  Next  place 

a  piece  of  plate  glass  carefully  cleaned  on  the  stand  between  A 
and  B,  and  at  right  angles  to  the  line  connecting  them.  To  make 
this  adjustment  remove  D,  and  place  the  eye  beyond  the  light. 
Then  turn  the  stand  until  A^  its  reflection  in  the  plate  glass,  and 
its  reflection  in  B  and  (7,  all  lie  in  the  same  straight  line.  Now 
set  the  disk  as  before,  record  the  mean  reading,  and  compute  the 
relative  intensities.  Increase  the  number  of  plates  one  at  a  time, 
and  compute  the  intensity  in  each  case.  This  number  divided  by 
that  when  no  plates  were  interposed,  gives  the  percentage  trans- 
mitted. 

This  same  apparatus  is  well  adapted  to  determine  the  amount 
of  light  transmitted  at  different  angles  of  incidence,  that  cut  off 
by  ground  glass,  the  effect  of  the  snuff  of  a  candle,  or  the  over- 
hanging portion  of  the  wick,  and  the  comparative  brilliancy  of 
the  edge  and  side  of  a  flat  flame.  In  the  last  case  it  is  only  nec- 
essary to  set  the  flame  so  that  it  shall  shine  edgewise,  first  into  B 
and  then  into  (7,  and  compare  the  position  of  the  disk  in  the  two 
cases.  It  will  thus  be  found  that  the  prevalent  impression  that 
flame  is  perfectly  transparent,  is  erroneous. 


184  DAYLIGHT     PHOTOMETER. 

68.    DAYLIGHT  PHOTOMETER. 

Apparatus.  In  Fig.  50,  AB  is  a  box  about  six  feet  long,  a  foot 
wide,  and  a  foot  and  a  half  high.  It  may  be  made  of  a  light 
wooden  frame  covered  with  black  paper  or  cloth.  A  circular  hole 
about  four  inches  in  diameter  is  cut  in  the  end  B,  and  covered 
with  blue  glazed  paper  with  the  white  side  out,  and  made  into  a 
Bunsen  disk  by  a  drop  of  melted  candle-wax  in  the  centre.  A 
long  wooden  rod  rests  on  the  bottom  of  the  box,  and  has  a  stand- 
ard wax  candle,  A^  in  a  spring  candle-stick  attached  to  one  end. 
The  distance  of  the  candle  from  the  disk  may  thus  be  varied  at 
will,  and  measured  by  a  scale  attached  to  the  rod.  The  box 
should  be  ventilated  by  suitable  holes  cut  in  it,  or  the  air  will 
become  so  impure  that  the  candle  will  not  burn  properly. 

Experiment.  This  instrument  is  intended  to  compare  the 
amount  of  light  in  different  portions  of  a  room,  or  its  brightness 

at  different  times. 
When  the  candle  is 
placed  at  a  distance 

0  .  from   the    photome- 

ter disk,  the   latter 
will  appear  dark  in 
Fi    50  the  centre,  while  by 

making    AB     very 

small,  so  that  the  strongest  light  shall  be  inside,  the  centre  will  be 
bright.  The  color  of  the  candl'e  flame  being  of  a  reddish  tint 
compared  with  daylight,  is  first  passed  through  the  blue  paper, 
which  thus  renders  the  colors  more  nearly  alike.  When  the  dis- 
tance of  the  candle  is  such  that  the  illumination  is  equal  on  both 
sides  of  the  disk,  the  spot  will  nearly  disappear,  and  unity  divided 
by  the  square  of  this  distance  gives  a  measure  of  the  comparative 
brightness  under  various  circumstances. 

An  excellent  experiment  with  this  instrument  is  to  measure  the 
fading  of  the  light  at  twilight.  Light  the  candle  and  place  it  at 
such  a  distance  from  the  disk  that  the  spot  shall  disappear,  as  in 
the  last  experiment.  As  the  light  diminishes,  the  distance  AB 
must  be  increased.  Take  readings  at  intervals  of  one  minute,  and 
construct  a  curve  with  ordinates  equal  to  one  divided  by  the 
square  of  this  distance,  and  abscissas  equal  to  the  time.  The 
amount  of  light  for  different  distances  of  the  sun  below  the  hori- 


BUNSEN    PHOTOMETER.  135 

zon  may  be  obtained  directly  from  this  curve.  In  the  same  way 
the  brightness  of  different  parts  of  the  laboratory  may  be  meas- 
ured, the  effect  of  drawing  the  window  curtains,  and  the  compara- 
tive brightness  of  clear  and  cloudy  days.  This  apparatus  was 
used  during  the  Total  Eclipse  of  1870,  to  measure  the  amount  of 
light  during  totality,  possessing  the  advantage  that  on  returning, 
the  precise  degree  of  darkness  could  be  reproduced  artificially. 
Comparisons  may  also  be  made  with  moonlight,  the  light  of  the 
aurora  or  other  similar  sources  of  light. 

69.    BUNSEN  PHOTOMETER. 

Apparatus.  A  photometer  room  forms  a  most  valuable  addition 
to  a  Physical  Laboratory,  both  on  account  of  the  great  variety  of 
original  investigations  which  may  easily  be  conducted  in  it  by 
students,  and  also  owing  to  the  practical  value  of  the  instrument, 
and  the  excellent  training  it  affords  in  the  use  of  various  forms  of  gas 
apparatus.  If  a  separate  room  cannot  be  obtained,  a  part  of  the 
laboratory  may  be  partitioned  off  by  paper  or  cloth  screens  black- 
ened on  the  interior,  so  as  to  leave  a  space  about  twelve  feet  long 
by  five  wide  and  eight  feet  high,  which  should  be  nearly  dark,  and 
supplied  with  some  means  of  ventilation.  In  this  is  a  table  ten 
feet  long,  a  foot  and  a  half  wide,  and  three  high,  and  over  its  cen- 
tre, at  a  height  of  five  feet  from  the  floor,  the  photometer  bar,  AB, 
Fig.  51,  is  placed.  The  latter  is  100  inches  in  length,  and  divided 
on  one  side  into  inches  and  tenths,  and  on  the  other  into  candle 
powers.  To  make  this  graduation,  calling  x  the  distance  from  one 
end  of  the  bar  in  inches,  and  C  the  corresponding  candle  power, 
we  have,  as  will  be  seen  below,  x2 :  (100  —  xY  =  C :  1,  or  x  = 

*J  C 

100  1  -L.  /  Q-  By  making  C  =  1,  2,  3,  etc.,  the  bar  may  be  grad- 
uated as  desired.  At  one  end  of  the  bar  is  placed  a  sperm  candle, 
-4,  supported  in  a  balance  for  determining  its  loss  of  weight  as  it 
burns.  The  best  form  is  that  invented  by  Prof.  F.  E.  Stimpson, 
on  the  principle  of  the  bent-lever  balance,  in  which  the  motion  of 
a  long  arm  over  a  scale  shows  the  number  of  grains  consumed. 

At  the  other  end  of  the  bar,  gas  is  admitted,  and  its  brightness 
when  burned  compared  with  that  of  the  candle.  The  pipe  sup- 
plying the  gas  passes  through  the  meter  F,  which  is  of  the  form 
known  as  the  wet  meter,  and  indicates  the  volume  to  one  thou- 
sandth of  a  foot.  To  read  it,  a  separate  burner  E  is  provided, 
supplied  with  gas,  which  does  not  pass  through  the  meter.  Of 
course  it  is  turned  down  when  setting  the  disk.  The  gas  passes 
from  the  meter  to  the  regulator  G,  by  which  the  pressure  is  ren- 
dered perfectly  constant.  This  consists  of  a  bell  resting  in  water, 
like  a  gas-holder,  with  a  long  conical  rod  attached  to  its  centre, 


136 


BUNSEN    PHOTOMETER. 


which,  when  raised,  cuts  off  the  supply  of  gas.  Whenever  the 
pressure  becomes  too  great,  the  bell  rises  and  reduces  the  flow  of 
gas,  while  too  small  a  pressure  makes  the  bell  descend  and  admit 
more  gas.  Beyond  the  regulator  a  f  is  inserted,  and  connected 
with  a  gauge  I,  which  gives  the  pressure.  In  its  simplest  form 
this  is  a  bent  tube  containing  water,  which  should  be  of  consider- 
able size,  if  accuracy  is  required.  Sometimes  a  floating  bell  is 
used,  which  rises  and  falls  as  the  pressure  varies,  and  moves  a  long 
index  over  a  graduated  scale.  The  gas  next  passes  to  the  burner 
B,  first  traversing  one  or  two  stopcocks,  H,  to  regulate  the  quan- 
tity consumed.  A  variety  of  burners  should  be  procured,  which 
may  be  used  in  turn  and  compared. 

A  slide  G  is  placed  on  the  photometer  bar,  carrying  a  Bunsen 
photometer  disk  (Experiment  67).  When  this  is  placed  between 
two  lights,  if  the  brightest  is  in  front,  the  small  circle  looks  light 
on  a  dark  ground,  if  the  brightest  is  behind,  it  appears  dark. 

A  clock  D,  marking  seconds,  is  also  needed  in  this  room,  the 
best  form  striking  a  bell  at  the  beginning  of  each  minute,  also  five 
seconds  before  it,  and  having  what  is  called  a  centre  seconds'  hand. 
It  is  often  convenient  to  have  a  separate  gas-pipe,  meter  and 
burner  at  A,  the  candle-end  of  the  photometer  bar,  and  to  have 
the  latter  arranged  so  that  it  can  be  swung  horizontally  to  it.  In 
fact,  in  almost  every  new  research  some  change  of  arrangement 
will  be  found  desirable,  and  the  apparatus  should  therefore  not  be 
fixed,  but  arranged  so  that  the  connections  can  be  easily  altered. 

Experiment.  To  measure  the  candle  power  of  burning  gas. 
The  law  in  the  State  of  Massachusetts  requires  that  the  gas  fur- 


Fig.  51. 

nished,  when  burnt  at  the  rate  of  five  feet  per  hour,  under  the 
most  favorable  circumstances,  shall  give  a  light  at  least  equal  to 


BUNSEN    PHOTOMETER.  137 

twelve  sperm  candles  (6  to  the  pound),  when  consuming  120 
grains  per  hour.  To  make  the  experiment  the  gas  and  candle  are 
burnt  at  opposite  ends  of  the  photometer  bar,  their  relative  inten- 
sities compared,  and  the  consumption  of  each  measured.  The 
amount  of  wax  burnt  is  measured  by  the  candle-balance  A.  This 
consists  of  a  sort  of  steelyard,  with  a  light  weight  or  rider  K, 
moving  over  its  longer  arm,  which  is  divided  so  as  to  give  grains. 
The  centre  of  gravity  of  the  beam  is  at  such  a  distance  from  the 
point  of  suspension  that  the  sensibility  shall  not  be  very  great,  and 
an  index  is  attached,  which  moves  over  a  scale,  each  of  whose 
divisions  corresponds  to  a  change  in  weight  of  one  grain. 

Attach  the  candle  to  the  shorter  end  of  the  beam  and  light  it ; 
set  the  rider  at  zero,  and  place  weights  in  the  scale-pan,  until  that 
end  of  the  balance  is  somewhat  the  heaviest.  Now  as  the  candle 
burns  it  becomes  lighter,  and  soon  begins  to  rise,  its  diminution 
grain  by  grain  being  shown  by  the  index  moving  over  the  scale. 
Light  the  gas  also  at  the  other  end  of  the  beam,  and  place  the 
photometer-disk  on  the  bar  ready  for  use.  Precisely  at  the  begin- 
ning of  a  minute,  as  given  by  the  clock,  read  the  gas-meter,  re- 
cording the  feet  and  thousandths,  also  the  position  of  the  index  of 
the  candle-balance.  The  observations  commonly  extend  over  five 
minutes,  and  in  this  time  10  grains  of  wax  should  be  consumed  • 
set  the  rider  therefore  at  10,  and  if  the  candle  is  burning  at  the 
standard  rate,  the  position  of  the  index  at  the  end  of  that  time 
will  be  the  same  as  at  the  beginning,  if  not,  the  difference  shows 
the  correction  to  be  applied. 

Next  measure  the  intensity  of  the  two  lights  by  the  photometer 
disk.  As  stated  above,  this  possesses  the  property  of  appearing 
light  on  a  dark  background,  or  the  contrary,  according  as  the 
brightest  light  is  in  front  of,  or  behind  it.  By  moving  it  back- 
wards and  forwards  therefore,  a  point  will  be  found  where  the 
spot  will  disappear  almost  completely,  owing  to  the  equality  of 
the  two  lights.  Read  its  position  by  the  graduation  and  record, 
then  move  it,  and  set  again  several  times.  At  the  end,  of  the  min- 
ute read  the  meter,  and  then  take  some  more  readings  of  the  disk. 
Try  also  setting  the  disk  so  that  the  spot  shall  be  first  slightly 
brighter,  and  then  equally  darker  than  the  adjacent  paper.  This 
is  called  taking  limits,  and  the  mean  gives  the  true  reading.  Pro- 


138  LAW    OF    REFLECTION. 

ceed  in  this  way  for  five  minutes,  reading  the  meter  at  the  end  of 
each  minute,  and  taking  two  or  three  intermediate  settings  ol  the 
disk.  At  the  end  of  the  time  read  the  index  of  the  candle-balance 
also. 

From  these  data  the  candle  power  may  be  computed  as  follows. 
The  consumption  of  the  candle  is  obtained  by  subtracting  the  first 
reading  of  the  index  from  the  last,  and  adding  the  difference  to 
10.  This  gives  the  consumption  in  5  minutes,  and  multiplying  it 
by  twelve  gives  (7,  the  number  of  grains  per  hour.  It  is,  however, 
safer  to  extend  the  reading  of  the  candle-balance  to  a  longer  time, 
as  fifteen  or  twenty  minutes,  to  diminish  the  errors.  The  con- 
sumption of  gas  per  minute  is  obtained  by  subtracting  each  read- 
ing of  the  meter  from  that  which  follows  it,  and  multiplying  by 
6Q  gives  6r,  the  rate  per  hour.  Call  I  the  ratio  of  the  two  lights, 
as  given  by  the  mean  of  the  readings  of  the  scale  attached  to  the 
bar,  and  apply  the  following  corrections.  First,  for  rate  of  candle^ 

it  is  assumed  that  the  light  is  proportional  to  the  consumption. 

r< 

Hence  the  corrected  candle  power  _Z7  :  L  =  C\  120,  L'  =  L  ~^- 

Again  it  is  assumed  that  the  light  of  the  gas  is  proportional  to  its 

C* 

consumption,  or  to  -JT-,  and  dividing  by  this  fraction  gives  what 

would  be  the  candle  power  if  just  5  feet  were  burned,  or  the  true 

5  (75 

candle  power  L"  =  L'~Q  \  hence  L"  =  L  •  J^Q  •  -g 

This  example  serves  to  show  how  the  photometer  is  ordinarily 
used,  but  it  may  be  applied  to  a  great  variety  of  investigations. 
For  instance,  different  burners  may  be  compared,  or  a  single 
burner  under  varying  consumption.  The  amount  of  light  cut  off 
by  plain  and  ground  glass  at  various  angles  may  be  measured, 
and  the  effect  of  changes  in  moisture,  in  temperature,  or  in 
.barometric  pressure  studied. 

70.    LAW  OF  REFLECTION. 

Apparatus.  In  Fig.  52  a  circle  divided  into  degrees  is  attached 
to  a  stand,  and  carries  two  arms  with  verniers,  or  simple  pointers, 
C  and  D.  The  first  is  attached  to  a  centre  plate,  which  carries 
a  vertical  mirror  placed  at  right  angles  to  BC.  This  mirror 
is  silvered  on  its  front  surface,  or  may  be  made  of  blackened  glass, 


B 


A 


ANGLES    OF    CRYSTALS.  139 

and  a  vertical  line  is  ruled  on  it,  which  is  brought  to  coincide  ex- 
actly with  the  centre  of  the  circle.  A  vertical  rod  or  needle  is 
attached  to  J9,  whose  reflection  in  B  is  to  be  observed  at  different 
angles  of  incidence.  A  is  a  piece  of  brass  with  a  small  hole 
pierced  in  it  to  look  through. 

Experiment.  Bring  O  in  line  with  B  and  A>  and  turn  it  so 
that  on  looking  through  the  latter  the  reflection  of  the  hole  may 
be  brought  to  the  centre  of  the  mir 
ror  and  bisected  by  the  line  marked 
on  it.  The  reading  of  the  index  C 
gives  the  zero,  or  starting  point. 
This  observation  should  be  repeated 
two  or  three  times,  dividing  the  de- 
grees into  tenths  by  the  eye.  Turn 

C  a  few  degrees  and  bring  D  into  such  a  position  that  the  reflec- 
tion of  its  needle  shall  coincide  with  the  line  on  C.  Now  the 
difference  in  reading  of  D  and  (7  will  equal  the  angle  of  incidence, 
and  the  difference  between  the  reading  of  C  and  the  zero  equals 
the  angle  of  reflection.  By  the  law  of  reflection  these  two  angles 
should  be  equal.  Repeat  this  observation  with  different  parts  of 
the  graduated  circle,  at  intervals  of  about  fifteen  or  twenty  de- 
grees. Small  deviations  from  the  law  serve  well  to  exemplify  the 
different  kinds  of  errors  of  observations.  Thus  if  the  needle  is 
not  exactly  on  the  line  connecting  its  index  with  (7,  a  constant 
error  will  be  introduced.  If  the  mirror  is  not  exactly  over  the 
centre  of  the  circle,  the  difference  will  vary  in  different  parts  of 
the  circle,  causing  a  periodic  error.  If  the  differences  between  the 
angles  of  incidence  and  reflection  are  sometimes  positive  and 
sometimes  negative,  they  are  probably  due  to  accidental  errors, 
such  as  errors  in  graduation,  in  reading,  unequal  fitting  of  the 
parts,  etc.  Finally,  if  a  single  observation  gives  a  large  error,  it  is 
probably  due  to  a  mistake,  or  totally  erroneous  reading. 

71.    ANGLES  OF  CRYSTALS. 

Apparatus.  The  instrument  most  commonly  employed  to 
measure  the  angles  of  crystals  is  Wollaston's  reflecting  goniometer, 
represented  in  Fig.  53.  A  is  a  vertical  circle  divided  into  degrees, 
and  turned  by  a  milled  head  B  through  any  given  angte,  which  is 
measured  by  a  vernier  C.  A  second  milled  head  D  is  attached  to 


140  ANGLES    OF    CRYSTALS. 

a  rod  passing  through  the  axis  of  this  circle  with  friction.  The 
crystal  is  fastened  by  wax  to  a  small  brass  plate  E,  bent  at  right 
angles  and  resting  in  the  split  end  of  the  pin  F.  By  this  it  may 
be  turned  horizontally,  and  the  joint  Gr  gives  a  vertical  motion. 
The  whole  is  mounted  on  a  stand,  which  should  be  placed  on  a 
table  opposite  the  window,  and  ten  or  twelve  feet  from  it.  A 
spring  stop  is  attached  to  the  stand,  and  a  pin  placed  in  the  grad- 
uated circle,  so  that  when  the  latter  is  turned  forward  it  cannot 
pass  the  0°,  or  180°  mark,  but  may  be  brought  by  the  milled  head 
exactly  to  this  point.  Several  crystals  to  be  measured  should  be 
provided,  some,  as  quartz,  galena,  alum  or  salt,  well  formed  and 
polished,  and  therefore  easily  measured,  and  others  of  greater  diffi- 
culty. The  best  material  with  which  to  attach  them  to  E  is  a 
little  beeswax. 

Experiment.  The  crystal  must  be  fastened  to  the  stand  in  such 
a  way  that  the  edge  to  be  measured  shall  lie  exactly  in  the  axis  of 
the  instrument  prolonged,  and  the  main  diffi- 
culty in  the  experiment  is  to  make  this  ad- 
justment with  accuracy.  Attach  the  crystal 
to  the  plate  E  by  a  little  piece  of  wax,  and 
adjust  the  edge  as  nearly  as  possible  by  the 
eye,  turning  it  horizontally  by  the  pin  F,  and 
vertically  around  the  joint  G.  It  is  thus 
brought  parallel  to  the  axis,  and  may  be  made 
to  coincide  with  it  by  sliding  the  plate  in  the 
Fig.  53.  pm  fi  Select  now  two  parallel  lines,  one  of 

which  may  be  a  bar  of  the  window,  and  the  other  the  further  edge 
of  the  table,  or  a  line  ruled  on  paper,  and  set  the  axis  of  the  instru- 
ment parallel  to  them.  On  bringing  the  eye  near  the  crystal  an 
image  of  the  window  will  be  seen  reflected  in  one  of  its  faces,  and 
by  turning  either  milled  head  the  image  of  the  bar  may  be  brought 
to  coincide  with  the  second  line.  If  they  are  not  parallel  it  shows 
that  the  face  is  not  parallel  to  the  axis  of  the  instrument,  and  the 
crystal  must  be  moved.  Do  the  same  with  the  other  face  to  be 
measured,  and  when  both  images  are  parallel  to  the  line  on  the 
table,  both  faces,  and  consequently  their  intersection,  are  parallel*  to 
the  axis.  This  adjustment  is  most  readily  made  by  placing  one 
face  as  nearly  as  possible  perpendicular  to  the  pin  F,  when  the 
image  in  this  face  may  be  rendered  parallel  by  turning  6r,  that  in 
the  other  by  turning  F. 


ANGLE    OF    PRISMS.  141 

Now  turn  B  until  the  circle  stops  at  180°,  and  turn  D  until  the 
image  in  the  further  face  of  the  crystal  coincides  exactly  with  the 
line  on  the  table.  Then  turn  B  in  the  other  direction  until  the 
second  image  coincides,  when  the  reading  of  the  vernier  will  give 
the  correct  angle.  Evidently  the  two  faces  are  in  turn  brought 
into  exactly  the  same  position,  and  the  angle  between  them  equals 
180°  minus  the  amount  through  which  the  circle  has  been  turned. 
It  is  sometimes  more  accurate,  though  a  little  more  troublesome,  to 
turn  the  crystal  into  any  position  by  J9,  and  bring  first  one  image 
to  coincide,  and  then  the  other.  180°  minus  the  difference  in  the 
readings  of  the  vernier  give  the  required  angle.  Try  this  with 
different  parts  of  the  circle.  Remove  the  crystal,  attach  it  a  sec- 
ond time  to  E,  and  see  if  the  same  result  is  attained  as  before. 
Repeat  until  readings  are  obtained  differing  from  each  other  but  a 
few  minutes.  Also  measure  some  of  the  crystals  less  highly 
polished.  An  excellent  test  of  the  work  is  to  measure  the 
angles  completely  around  a  crystal,  and  see  if  their  sum  equals 
180° (n  —  2),  in  which  n  is  their  number. 

72.     ANGLE  OF  PKISMS. 

Apparatus.  One  of  the  most  valuable  instruments  in  a  Physi- 
cal Laboratory  is  the  Optical  Circle,  or  Babinet's  goniometer. 
This  instrument  may  be  used  as  a  goniometer  for  measuring  the 
angles  of  crystals,  to  find  the  index  of  refraction  of  liquids  or 
solids,  to  study  dispersion,  or,  as  a  spectrometer,  to  measure  wave- 
le/ngths.  It  is  therefore  often  desirable  to  duplicate  it,  or  perhaps 
better,  to  procure  one  large  and  very  accurate  instrument,  and 
others  of  smaller  size  for  work  requiring  less  precision. 

This  instrument,  Fig.  54,  consists  of  a  graduated  circle  on  a 
stand,  with  two  telescopes,  A  and  B,  attached  to  it.  A  is  the  col- 
limator,  or  a  telescope  in  which  the 
eye-piece  is  replaced  by  a  fine  slit, 
whose  width  may  be  varied  by  a  screw- 
resting  against  a  spring,  and  whose 
distance  from  the  object-glass  may  be 
altered  by  a  sliding  tube  with  a  rack 
and  pinion.  This  telescope  is  at- 
tached permanently  to  the  stand,  while 
B,  which  is  a  common  telescope  with 
cross  hairs  in  its  focus,  is  fastened  to 
an  arm  revolving  around  the  centre  of  Fi£-  54- 

the  graduated  circle.      It  may  be  held  in   any  desired  position 
by  a  clamp  D,  moved  slowly  by  a  tangent  screw  E,  and  the  angle 


142  ANGLE     OF    PRISMS. 

through  which  it  has  been  turned,  accurately  measured  by  a 
vernier.  To  eliminate  errors  of  eccentricity  a  second  vernier  is 
sometimes  placed  opposite  the  first,  in  which  case  the  mean  of 
their  readings  is  always  employed.  For  great  accuracy  a  spider- 
line  micrometer  should  be  attached  to  J5  to  measure  small  angles, 
as  will  be  described  more  in  detail  in  Experiment  77.  It  is  often 
convenient  to  have  both  telescopes  mounted  on  conical  bearings 
so  that  they  may  be  turned  away  from  the  centre  of  the  circle 
when  desired.  They  should  also  be  supported  in  such  a  way  that 
one  end  of  each  may  be  raised  or  lowered  a  little,  so  as  to  bring 
their  axes  perpendicular  to  that  of  the  instrument.  This  is  most 
readily  accomplished  by  placing  an  adjusting  screw  under  one  of 
the  Y's  carrying  them.  C  is  a  small  circular  stand  on  which  prisms 
may  be  placed,  and  which  may  be  turned  around  the  centre  of  the 
circle  and  clamped  in  any  position.  Sometimes  an  arm  and  ver- 
nier is  attached  to  measure  its  angular  motion,  but  this  is  not  ab- 
solutely necessary.  Its  principal  use  is  to  measure  the  angle  of 
crystals,  and  by  it  the  law  of  reflection  may  also  be  proved  with 
great  accuracy.  The  graduated  circle  is  sometimes  made  to  re- 
volve, and  the  angle  measured  by  one  or  more  fixed  verniers. 

The  whole  is  commonly  mounted  on  a  tripod  with  levelling 
screws,  as  shown  in  the  figure.  These  are  ornamental  rather 
than  useful,  however,  as  in  common  experiments  it  makes  no  dif- 
ference, except  in  appearance,  if  the  circle  is  not  properly  lev- 
elled. In  any  case,  except  to  raise  or  lower  the  instrument,  only 

two  screws  are  needed,  and  the  third  may  be 

—        replaced  by  a  fixed  point.     In  this,  as  in  all 

^  instruments  mounted  on  three  legs,  the  best 

form  of  support  is  that  represented  in  Fig.  55. 

One  leg  rests  in  a  conical  hole  A,  a  second 

in  a  wedge-shaped  groove  B,  and  the  third  on 
9  a  plane  surface   C.    A  fixes  the  position  of 

the    tripod,   which   is   prevented   from    turn- 
Fig  55  *n&  ky  the  groove  .7?,  while  if  the  three  legs 

change  their  relative  positions,  IB  can  slide 
back  and  forth  in  its  groove,  and  C  move  freely  over  the  plane 
surface.  If  instead,  three  conical  holes  were  used,  and  these  were 
not  precisely  in  the  right  position,  or  the  distance  of  the  legs 
varied  with  changes  of  temperature,  the  whole  instrument  might 
be  so  strained  as  to  introduce  serious  errors  in  the  graduated 
circle.  This  instrument  should  be  placed  near  the  window  so  that 
sunlight  may  be  reflected  through  it  by  means  of  a  mirror,  or  if 
preferred,  the  light  from  an  Argand  or  Bunsen  burner  employed. 
One  or  more  flint  glass  prisms  are  also  needed,  all  three  of  whose 
faces  should  be  polished  and  inclined  at  angles  of  60°. 

Experiment.  The  following  adjustment  must  always  be  made 
when  the  optical  circle  is  used.  Draw  out  the  eye-piece  of  .Z?, 


ANGLE     OF    PRISMS.  143 

Fig.  54,  until  the  cross-hairs  are  seen  with  perfect  distinctness. 
Then  turn  the  telescope  towards  some  distant' object  and  focus  it, 
moving  both  eye-piece  and  cross-hairs.  Now  both  the  object  and 
cross-hairs  should  be  perfectly  distinct,  and  not  change  their  rela- 
tive positions  as  the  eye  is  moved  from  side  to  side  so  as  to  look 
through  different  portions  of  the  eye-lens.  Sometimes  the  objective 
alone  moves,  and  sometimes  the  distance  is  permanently  fixed,  so 
that  it  is  in  adjustment  for  parallel  rays.  Now  turn  the  two  teles- 
copes towards  each  other  and  illuminate  the  slit  either  by  placing 
an  Argand  burner  behind  it,  or  reflecting  the  light  of  the  sky 
through  it  by  means  of  a  mirror.  On  looking  through  the  ob- 
serving telescope  an  image  of  the  slit  will  now  be  visible.  Focus 
it,  moving  it  towards  or  from  its  objective,  when  its  distance  will 
equal  the  principal  focal  distance  of  the  collimator,  and  the  beam 
of  light  between  the  two  telescopes  will  be  parallel,  or  as  if  coming 
from  a  slit  placed  at  a  very  great  distance.  Bring  the  image  of 
the  slit  to  coincide  exactly  with  the  vertical  cross-hairs  in  B  by 
the  tangent  screw,  first  champing  the  telescope.  If  it  is  not  verti- 
cal the  slit  may  be  turned,  and  if  it  is  too  high  or  too  low  it  should 
be  brought  to  the  centre  of  the  field  by  raising  or  lowering  one 
end  of  one  of  the  telescopes,  as  described  above.  Having  ren- 
dered the  coincidence  exact,  read  the  vernier  and  repeat  the 
setting  two  or  three  times,  as  it  gives  the  zero  from  which  most  of 
the  following  measurements  are  made. 

To  measure  the  angle  of  a  prism,  stand  it  on  the  centre-plate 
with  its  edges  vertical,  and  with  the  faces  whose  angle  is  to  be 
determined  about  equally  inclined  to  the  axis  of  the  collimator. 
To  eliminate  parallax  in  case  the  telescopes  have  not  been  accu- 
rately focussed  for  parallel  rays,  it  is  better  to  place  the  edge  of 
the  prism  over  the  centre  of  the  graduated 
circle.  To  'prevent  motion  of  the  prism 
when  IB  is  turned,  C  should  be  clamped. 
Let  AJ3C,  Fig.  56,  represent  the  prism 
whose  angle  A  is  to  be  measured,  and  DDf 
the  axis  of  the  collimator  prolonged.  Turn 
the  observing  telescope  into  the  position 
AF,  when  an  image  of  the  slit  will  be  seen 
on  looking  through.  Bring  it  to  coincide 


144  ANGLE    OF    PRISMS. 

with  the  cross-hairs  by  the  tangent-screw,  first  clamping  the  teles- 
cope, and  read  the  vernier.  Then  turn  the  telescope  into  the 
position  AE  and  set  again.  The  difference  in  the  readings  di- 
vided by  two,  equals  the  angle  of  the  prism.  For  D 'A  C  equals 
90°  —  the  angle  of  incidence,  and  FAC,  90°  —  the  angle  of  re- 
flection ;  hence  they  are  equal.  In  the  same  way,  EAB  =  D'  AB, 
or  FA  C  +  EAB  =  BA  C,  and  FAE  —  IB  A  C.  Move  the  prism 
a  little,  repeat  the  measurement,  and  see  if  the  same  result  is 
obtained  as  before.  Determine  in  the  same  way  the  three  angles 
of  the  prism,  and  their  sum  should  equal  180°.  If  either  of  the 
reflected  images  of  the  slit  is  too  high  or  too  low,  the  base  of 
the  prism  is  not  perpendicular  to  the  edges.  In  this  case  it  must 
be  adjusted  by  placing  pieces  of  paper,  or  tinfoil,  under  one  or  two 
of  its  corners,  until  both  images  are  in  the  centre.  If  either  is 
out  of  focus  when  the  telescopes  have  been  adjusted  for  parallel 
rays  the  reflecting  surface  is  curved  instead  of  plane,  while  a  dis- 
tortion of  the  image  shows  that  the  surface  is  irregular.  In  either 
case,  an  accurate  measurement  is  impossible,  since  the  angle  will 
vary  for  different  parts  of  each  face. 

By  the  plan  just  described,  the  angle  of  a  prism  may  be  found 
if,  as  is  often  the  case,  the  centre  plate  has  no  vernier  attached  to  it. 
With  such  a  vernier,  however,  the  angle  may  be  determined  more 
readily,  as  follows.  Set  the  telescopes  nearly  at  right  angles,  and 
stand  the  prism  on  the  centre-plate,  as  before,  with  its  faces  verti- 
cal, and  the  edge  to  be  measured  over  the  axis  of  the  instrument. 
Turn  the  centre-plate  until  one  of  the  faces  is  equally  inclined  to 
the  axes  of  both  telescopes,  when  the  image  of  the  slit  reflected  in 
this  face  will  be  seen  in  the  field  on  looking  through  the  observing 
telescope.  Bring  it  to  coincide  with  the  cross-hairs  by  the  clamp 
and  tangent-screw,  and  read  the  vernier.  Turn  the  centre-plate, 
taking  great  care  not  to  disturb  the  position  of  the  prism  on  it, 
until  the  image  reflected  in  the  other  face  coincides  with  the  cross- 
hairs. 180°  minus  the  difference  in  the  readings  of  the  vernier 
gives  the  angle  of  the  prism.  Repeat  as  before,  and  also  measure 
the  three  angles  and  see  if  their  sum  equals  180°. 

When  a  vernier  is  attached  to  the  centre-plate  this  instrument 
serves  to  prove  the  law  of  reflection  with  great  exactness.  For 
this  purpose  it  is  only  necessary  to  turn  the  centre-plate  into  vari- 


LAW    OF    REFRACTION.     I. 


145 


ous  positions,  bring  the  reflection  of  the  slit  to  coincide-  with  the 
cross-hairs  of  the  observing  telescope,  and  read  the  verniers  at- 
tached to  each ;  or  in  fact,  to  repeat  Experiment  70,  replacing  the 
sight-hole  by  the  slit  and  collimator,  and  the  needle  by  the  ob- 
serving telescope. 

73.    LAW  OF  REFRACTION.    I. 

Apparatus.  In  Fig.  57,  DBCG  is  a  tank,  like  that  of  an  aqua- 
rium, with  the  side  BD  of  glass.  Two  horizontal  scales  are  at- 
tached to  CG,  one  over  the  other,  and  the  tank  filled  so  that  one 
shall  be  seen  above,  the  other  below,  the  liquid.  A  is  a  plate  of 
brass  with  a  vertical  slit  in  it,  larger  above,  and  tapering  to  a 
point.  It  is  used  as  a  sight,  and  is  placed  at  a  distance  AB  equal 
to  BC.  A  small  plumb-line  may  be  hung  in  front  of  DB  to  serve 
as  an  index.  A  tank  without  glass  sides  may  be  employed  in- 
stead, by  regarding  Fig.  57  as  a  vertical  instead  of  a  horizontal 
section,  and'placing  one  scale  at  the  surface  of  the  water,  BD,  the 
other  at  the  bottom,  CGr.  The  divisions  of  the  upper  scale  should 
then  be  one  half  those  of  the  lower. 

Experiment.  On  looking  through  A*ihe  lower  scale  will  be 
seen  through  the  water,  the  upper  through  the  air  only.  The 
divisions  of  the  former  will  therefore, 
by  refraction,  appear  larger  than  those 
of  the  other,  and  from  the  amount  of 
this  increase  the  law  of  refraction  may 
be  deduced.  Placing  the  plumb-line  at 
D,  and  looking  through  A,  it  will  be 
seen  projected  on  the  upper  scale  at 
6r,  but  on  the  lower,  owing  to  the  bend-  ^,: 
ing  of  the  ray  at  the  surface  BD,  at  !?• 
If  placed  at  B,  however,  the  reading  on 
both  scales  will  be  the  same,  since  the  incidence  being  normal 
there  is  no  bending  of  the  ray.  To  find  this  point,  read  both 
scales,  and  if  the  reading  of  the  upper  scale  is  the  greatest,  move 
B  to  the  right,  otherwise  to  the  left,  until  both  read  alike.  The 
object  of  the  varying  width  of  the  slit  is  to  read  approximately 
through  the  upper  part,  and  then  lowering  the  eye  to  eliminate 
parallax,  and  read  more  exactly  by  the  lower  portion.  Move  the 
plumb-line  a  short  distance,  read  both  scales  again,  and  thus  take 
ten  or  fifteen  readings  between  B  and  D. 
10 


Fig.  57. 


146  LAW    OF    REFRACTION.    II. 

ISTow  from  these  readings  to  prove  that  "the  ratio  of  the  sines  of 
the  angles  of  incidence  and  refraction  is  always  constant  and 
equal  to  the  index  of  refraction.  In  the  figure,  the  angle  of  inci- 
dence equals  90°  —  ADB,  and  the  angle  of  refraction  EDF. 
To  find  these  angles,  subtract  the  reading  of  C  from  that  of  £, 
which  gives  GC,  and  dividing  by  two  gives  EG,  or  CE,  since 
ABD  equals  DEG.  In  the  same  way,  subtracting  the  reading  of 
C  from  that  of  F  gives  CF,  and  subtracting  CE,  found  above, 
gives  FE.  Dividing  EG  and  EF  by  DE  gives  the  tangents  of 
the  angles  of  incidence  and  reflection,  from  which  these  angles 
may  be  found.  The  ratio  of  their  sines,  or  the  difference  of  their 
logarithmic  sines,  should  then  be  constant,  and  give  the  index  of 
refraction.  This  in  the  case  of  water  equals  1.33.  If  preferred, 
AB  need  not  equal  BC,  but  it  is  more  convenient  to  have  them 
both  equal  to  some  simple  number,  as  10  inches. 

74.     LAW  OF  REFRACTION.    II. 

Apparatus.  The  instrument  represented  in  Fig.  52  may,  by  a 
slight  change,  be  employed  to  prove  the  law  of  refraction.  The 
graduated  circle  is  mounted  vertically,  the  needle  D  replaced  by 
a  narrow  slit,  the  mirror  B  removed,  and  a  test-tube  attached 
to  the  index  C.  This  test-tube  is  held  by  a  strip  of  brass,  whose 
top  is  just  on  a  level  with  the  centre  of  the  circle.  If,  then,  it  is 
filled  with  water,  so  that  the  bottom  of  the  meniscus  is  just  above 
the  brass,  the  top  of  the  water  will  be  just  on  a  level  with  the 
centre  of  the  circle,  even  if  the  tube  is  inclined.  To  mark  the 
direction  of  the  ray  in  the  liquid,  two  diaphragms  with  slits  in 
them  are  placed  in  the  tube,  one  at  the  bottom,  the  other  in  the 
middle.  A  piece  of  white  paper,  or  a  mirror,  should  be  placed 
below  to  reflect  light  up  through  the  tube,  and  the  whole  should 
be  mounted  on  levelling-screws,  and  placed  in  a  good  light  oppo- 
site the  window. 

Experiment.  By  the  following  method,  the  law  of  refraction, 
that  the  ratio  of  the  sines  of  the  angles  of  incidence  and  refraction 
is  a  constant,  may  be  proved  more  directly  than  in  the  last  experi- 
ment. Set  the  index  C  at  90°,  so  that  the  test-tube  shall  be 
vertical,  and  move  the  other  index  carrying  the  slit,  to  270°. 
The  three  slits  will  now  be  in  the  same  vertical  line,  and  on  look- 
ing through  the  upper  one,  light  will  be  seen  through  the  other 
two.  If  not,  they  must  be  brought  into  this  position  by  moving 
the  test-tube.  Fill  the  latter  with  water  until  light  is  just  visible 


INDEX    OF  REFRACTION.  147 

above  the  brass  strip.  If  now  the  test-tube  is  inclined  by  moving 
its  index,  the  other  index  must  be  moved  by  a  larger  amount  to 
bring  the  three  slits  again  apparently  in  line,  owing  to  the  refrac- 
tion at  the  surface  of  the  water.  And  the  angles  through  which 
these  indices  have  been  moved  will  equal  the  angles  of  refraction 
and  incidence,  respectively.  Before  making  the  measurement, 
however,  the  line  connecting  the  indices  in  their  first  position 
must  be  brought  at  right  angles  to  the  surface  of  the  water.  For 
this  purpose  turn  the  upper  index  70°  or  80°,  or  as  far  as  readings 
can  be  conveniently  taken,  and  turn  the  lower  index  until  light 
passes  through  the  three  slits.  Read  its  position  and  turn  each 
index  as  much  on  the  other  side.  If  light  is  again  visible  through 
the  slits,  no  further  correction  is  necessary.  If  not,  turn  the  level- 
ling screws  through  one  half  the  distance  required  to  bring  them 
apparently  in  line.  By  repeating  this  correction  the  adjustment 
may  be  made  exact.  Then  take  a  number  of  readings  of  the  two 
indices,  moving  the  upper  one  a  few  degrees,  and  turning  the 
lower  one  until  light  is  visible  through  the  slits.  Subtracting  from 
these  readings  90°  and  270°,  gives  the  angles  of  incidence  and 
refraction.  The  difference  of  the  logarithm  of  their  natural 
sines  will  equal  the  logarithm  of  the  index  of  refraction. 

75.    INDEX  OF  REFRACTION. 

Apparatus.  The  instrument  devised  by  Wollaston  to  measure 
the  index  of  refraction  is  represented  in  Fig.  58.  A  cube  or  right- 
angled  prism  of  glass  A,  rests  on  a  plate  of  glass  in  which  a  slight 
depression  has  been  ground.  The  system  of  jointed  bars  JBC,  CE 
and  DF,  is  attached  to  this,  so  that  when  C  is  raised,  F  and  B  slide 
towards  E.  BC  is  exactly  10  inches  long,  and  carries  two  sights 
through  which  A  maybe  viewed.  EC  equals  10  inches  multiplied 
by  the  index  of  refraction  of  the  prism,  and  DC  =.  DE  —  DF\ 
hence  E  is  always  vertically  under  (?,  because  if  a  circle  is  de- 
scribed with  centre  D  and  radius  D  (7,  CFE  will  be  a  right-angle, 
being  inscribed  in  a  semicircle.  EF  is  divided  into  inches,  but 
the  graduation  need  extend  only  from  about  12  to  15|  inches,  if 
the  index  of  refraction  is  1.55.  Bottles  containing  several  liquids 
to  be  measured  are  also  needed,  as  water,  alcohol,  turpentine  and 
various  oils,  and  some  solids  with  polished  surfaces,  as  mica,  quartz, 
and  marble. 

Experiment.     Place  a  drop  of  water  in  the  hollow  under  the 
prism  and  raise  G.     On  looking  through  the  sights  the  spot  where 


148  CHEMICAL    SPECTROSCOPE. 

the  prism  rests  on  the  drop  is  at  first  bright,  but  after  passing  a 
certain  position,  becomes  dark.  The  bounding  line  is  marked  by 

colors,  and  bringing  it  into  the  mid- 
dle of  the  field  of  view  and  reading 
F  should  give  13.35,  which  divided 
by  10,  or  1.335,  is  the  index  of  re-^ 
fraction  of  water.  The  explanation 
is,  that  when  C  is  low,  total  reflection 

takes  place,  and  the  spot  appears  bright ;  but  when  raised  the 
light  is  mostly  transmitted.  The  colored  line  appears  at  the 
angle  of  total  reflection,  in  which  case  the  sine  of  the  angle 
of  incidence  equals  the  ratio  of  the  indices  of  refraction  of 
the  two  media.  Call  n  and  nf  the  indices  of  the  glass  and 
given  liquid,  i  the  angle  of  incidence  of  the  ray  through  the  sights 
upon  the  prism,  r  its  angle  of  refraction,  and  90°  -  -  r  its  angle  of 
incidence  on  the  reflecting  surface  of  glass  and  liquid.  Then  be- 
ing at  the  angle  of  total  reflection,  sin  (90°  —  r)  =  cos  r  =  —  > 

orn'  =  n  cos  r,  and  the  problem  is  to  prove  that  this  equals  EF, 
calling  CB,  or  10  inches,  equal  to  unity.  Now  sin  CBE :  sin  CEB 
=  CE  :  GB=  n  :  1.  But  CBE  —  i,  hence  GEB  —  r.  Again, 
since  F\%  vertically  under  (?,  EF  =  CE  cos  CEF  =  n  cos  r,  or 
equals  nr. 

Wipe  the  drop  of  water  carefully  from  under  A^  and  replace  it 
by  other  liquids  in  turn.  Next  measure  the  indices  of  one  or 
more  solids,  by  cementing  them  to  the  glass  by  a  drop  of  some 
liquid  of  higher  index,  as  balsam  of  tolu. 

76.    CHEMICAL  SPECTROSCOPE. 

Apparatus.  A  common  chemical  spectroscope  with  one  prism, 
and  a  photographed  scale  to  measure  the  position  of  the  lines.  Two 
Bunsen  burners,  an  Argand  burner,  some  platinum  wires  sealed 
into  the  ends  of  glass  tubes,  and  two  stands  to  hold  them  a  little 
lower  than  the  slit  of  the  spectroscope.  A  dozen  small  vials  are 
set  in  a  stand  formed  by  boring  holes  in  a  block  of  wood,  and  filled 
with  the  substances  to  be  tested.  Part  contain  salts  of  sodium,  lith- 
ium, strontium,  calcium,  barium,  thallium,  etc.,  and  the  remainder 
labelled  A,  B,  C,  etc.,  contain  mixtures  of  these  substances.  In 
Fig.  59,  the  light  enters  the  instrument  through  a  slit  in  the  end 
of  the  tube  B.  At  the  other  end  of  this  tube  is  a  lens,  whose 


CHEMICAL    SPECTROSCOPE.  149 

focus  equals  the  length  of  the  tube,  so  that  the  rays  emerging  from 
it  are  parallel,  that  is,  the  same  effect  is  produced  as  if  the  slit 
were  placed  at  an  infinite  distance.  The  width  of  the  slit  is  varied 
by  means  of  a  screw  acting  against  a  spring,  so  that  more  or  less 
light  may  be  admitted.  The  rays  next  encounter  the  prism  Z>,  by 
which  they  are  refracted,  the  different  colors  being  bent  unequally, 
and  then  enter  the  observing  telescope  A.  A  series  of  images  of 
the  slit  are  thus  produced,  one  for  each  color,  forming  a  continuous 
band  of  colored  light,  red  at  one  end,  and  violet  at  the  other.  To 
measure  the  position  of  the  different  parts  of  this  spectrum,  a 
third  tube,  (7,  is  employed,  which  carries  at  its  outer  end  a  fine 
scale  photographed  on  glass,  and  the  rays  from  it  are  rendered 
parallel  by  a  lens,  as  in  the  case  of  J2.  G  is  set  at  such  an  angle 
that  the  image  of  the  scale  reflected  in  the  face  of  the  prism  is 
visible  through  the  observing  telescope  A,  at  the  same  time  as  the 
spectrum.  To  exclude  the  stray  light,  D  must  either  be  enclosed 
in  a  box,  or  covered  with  a  black  cloth. 

Experiment.     Turn  B  towards  the  window,  or  better,  reflect  a 
ray  of  sunlight  through  it,  and  nearly  close  the  slit.     A  brilliant 
band  of  color  becomes  visible  through  A, 
red  and  yellow  at  one  end,  and  blue  and 
violet  at  the  other.     Slide  the  eye-piece 
of  A  in  or  out,  until  the  edges  are  sharply 
marked,  when  fine  lines  will  be  seen  at 
right  angles  to  its  length.     These   must 
be  focussed  with  care,  noticing  that  with 

a  wide  slit  they  disappear  entirely,  while  with  a  very  narrow  one 
they  are  obscured  by  other  lines  at  right  angles  to  them,  due  to 
irregularities  in  the  slit,  or  dust  on  its  edges.  Light  the  Argand 
burner  and  place  it  near  (?,  drawing  the  scale  in  or  out,  until  its 
image  is  distinctly  visible  through  A.  This  adjustment  is  aided 
by  closing  the  slit,  or  covering  it  up.  Both  the  lines  and  scale 
should  no\v  be  distinctly  visible,  and  the  position  of  the  former 
may  be  accurately  determined  by  the  latter.  Record  in  this  way 
the  position  of  a  number  of  the  more  prominent  lines  in.  the  solar 
spectrum. 

Tarn  the  slit  away  from  the  window,  so  that  the  field  of  view 
shall  be  dark,  and  light  one  of  the  Bunsen  burners,  placing  it 
opposite  the  slit,  and  three  or  four  inches  distant.  Heat  one  of 
the  platinum  wires  in  it,  until  it  ceases  to  color  the  flame.  Then 
dip  it  in  the  vial  containing  soda,  and  place  it  on  its  stand  in  the 


150 


CHEMICAL   SPECTROSCOPE. 


flame.  On  looking  through  A,  a  brilliant  yellow  line  is  visible, 
which,  with  a  more  powerful  instrument,  is  seen  to  be  double,  or 
to  consist  of  two  fine  lines  very  near  together.  This  line  is  very 
characteristic,  and  by  it  an  almost  infinitesimal  amount  of  soda 
may  be  detected.  In  fact,  it  shows  itself  in  all  ordinary  sub- 
stances. Record  the  position  of  this  line,  burn  the  wire  •  clean, 
and  repeat  with  lithia  and  the  other  substances.  Most  of  them 
give  several  lines,  which  sometimes  become  more  visible  after  the 
wire  has  remained  in  the  flame  for  some  time.  Thus  strontia 
gives  a  blue  line,  a  strongly  marked  orange  line,  and  six  red  lines, 
all  of  whose  positions  should  be  accurately  recorded.  To  save 
time,  it  is  best  to  use  two  wires,  observing  with  one,  while  the 
other  is  being  cleaned  by  heating  it  with  the  second  Bunsen 
burner.  It  may  sometimes  be  cleaned  more  quickly  by  heating  it 
to  redness,  and  dipping  it  in  cold  water,  or  by  crushing  the  bead 
formed  with  a  pair  of  flat-nosed  pliers ;  but  care  must  then  be 
taken  not  to  break  the  wire.  If  the  salt  will  not  adhere  to  the 
wire,  the  latter  may  be  moistened  with  distilled  water,  or  a  loop 
made  in  its  end.  Having  measured,  and  become  familiar  with 
these  spectra^  try  some  of  the  contents  of  vial  A.  See  first  if  its 
ingredients  can  be  recognized  from  its  spectrum  by  the  eye,  and 
then  measure  all  the  lines,  and  compare  with  the  measurements 
previously  taken.  The  substances  present  may  be  found  by  the 
law  that  the  spectrum  of  a  mixture  of  any  bodies  contains  all  the 
lines  they  give  separately. 

All  measurements  made  with  a  spectroscope  must  be  reduced  to 
some  common  scale,  in  order  that  they  may  be  of  real  value,  as 
the  scales  attached  to  these  instruments  are  quite  arbitrary,  no 
two  being  alike.  To  make  such  a  reduction,  the  lines  measured  in 
the  solar  spectrum  must  first  be  identified.  The  table  given  in 
Expeiiment  No.  77,  on  p.  152,  may  be  employed  for  this  purpose. 
If  the  light  used  is  only  that  of  the  sky,  instead  of  sunlight,  the 
visible  spectrum  will  only  extend  from  near  B  to  G.  Having 
identified  the  intermediate  lines,  construct  points  with  ordinates 
equal  to  their  observed  position,  and  abscissas  equal  to  their  wave- 
lengths. A  curve  is  thus  obtained,  from  which  any  readings  of 
the  spectroscope  may  be  reduced  to  wave-lengths  by  inspection. 
Apply  it  to  the  lines  of  the  metals  measured  above. 


SOLAR    SPECTROSCOPE.  151 

77.     SOLAR  SPECTROSCOPE. 

Apparatus.  The  optical  circle,  a  60°  prism  of  dense  flint  glass, 
and  a  mirror  capable  of  turning  horizontally  or  vertically,  by 
which  a  ray  of  sunlight  may  be  reflected  in  any  desired  direction. 
This  is  accomplished  more  perfectly  by  a  heliostat,  in  which  the 
apparent  motion  of  the  sun  is  corrected  by  moving  the  mirror  by 
clockwork.  The  instrument  should  be  placed  near  a  window  into 
which  the  sun  is  shining,  or  if  the  day  is  cloudy,  the  experiment 
may  first  be  performed  with  a  Bunsen  burner  and  a  wire  on 
which  is  a  little  borax,  and  afterwards  concluded  by  the  aid  of 
sunlight. 

Experiment.  Adjust  the  optical  circle  as  'described  in  Experi- 
ment 72,  so  that  both  telescopes  shall  be  focussed  for  parallel  rays, 
and  when  placed  opposite  each  other  the  cross-hairs  and  slit  shall 
be  distinctly  visible.  Bring  them  to  coincide,  and  read  the  ver- 
nier. Turn  the  observing  telescope  about  45°,  and  place  the  prism 
on  the  centre  plate,  so  that  its  back  shall  be  equally  inclined  to 
the  axes  of  the  telescope  and  collimator,  as  at  D,  Fig.  59.  Place 
the  mirror  in  the  sunlight,  and  turn  the  collimator  towards  it,  and 
distant  only  a  few  inches.  Nearly  close  the  slit,  and  reflect  the 
light  through  it  by  turning  the  mirror.  On  moving  the  telescope 
to  one  side  or  the  other,  if  necessary,  a  brilliant  spectrum  will  be 
visible,  any  part  of  which  may  be  brought  to  coincide  with  the 
cross-hairs,  and  its  position  determined  by  the  vernier. 

To  obtain  the  best  results,  the  position  of  the  mirror  must  be 
accurately  adjusted.  This  may  be  done  in  two  ways.  Most  sim- 
ply by  opening  the  slit  wide,  when  the  position  of  the  beam  of 
sunlight  may  be  seen  in  its  passage  through  the  object-glass  of 
the  collimator,  forming  a  bright  spot  on  it.  The  mirror  should 
then  be  turned  until  this  spot  falls  in  the  centre.  Holding  a 
sheet  of  thin  paper  against  the  object-glass  renders  the  spot  more 
visible.  The  slit  must  be  nearly  closed  before  looking  through 
the  telescope,  or  the  eye  may  be  injured  by  the  intense  light.  A 
more  accurate  method  of  adjustment  is  to  remove  the  eye-piece 
and  look  through  the  tube,  when  an  image  of  objects  reflected  in 
the  mirror  will  be  faintly  visible.  Turn  the  mirror  until  the  im- 
age of  the  sun  falls  in  the  centre  of  the  object-glass  and  the 
light  will  then  pass  through  the  axes  of  both  telescopes.  At  the 
same  time  the  prism  should  be  placed  so  that  it  shall  cover  as 


152 


SOLAR    SPECTROSCOPE. 


much  of  the  object-glass  as  possible.  If  no  light  is  seen,  even 
if  the  slit  is  opened  wide,  probably  the  telescopes  are  not  set  at 
the  right  angle.  The  brilliancy  of  the  image  depends  on  the 
width  of  the  slit,  and  when  the  latter  is  very  narrow,  the  image 
of  the  sun  will  widen  out  by  diffraction.  Having  set  the  mirror 
correctly,  it  will  remain  right  only  for  a  few  minutes,  owing  to 
the  apparent  motion  of  the  sun,  and  hence  must  be  readjusted 
every  little  while.  This  may  commonly  be  done  with  sufficient 
accuracy  without  removing  the  eye-piece.  Much  trouble  may  be 
saved  by  noting  the  point  on  the  opposite  wall  where  the  reflected 
beam  falls,  and  resetting  the  mirror  by  this.  Or,  a  small  mirror 
may  be  attached,  and  the  direction  of  its  reflected  beam  noticed. 
If  the  shadow  of  a  window-sash  falls  on  the  mirror,  move  the  lat- 
ter across  it,  so  that  the  further  motion  of  the  sun  may  separate 
them  instead  of  again  bringing  them  together. 

Next  bring  the  prism  to  the  minimum  of  deviation,  that  is,  so 
that  its  back  shall  be  equally  inclined  to  both  telescopes.  Turn 
the  prism  while  looking  through  the  telescope,  and  the  spectrum 
will  ]be  seen  to  move  a  certain  distance  toward  the  red  end,  and 
then  return.  As  a  considerable  motion  of  the  prism  corresponds 
to  but  a  slight  motion  of  spectrum,  this  point  may  be  found  with 
sufficient  accuracy  by  the  hand  alone.  Now  focussing  the  teles- 
cope with  care,  the  spectrum  will  be  seen  to  be  traversed  by  a  mul- 
titude of  fine  vertical  lines  known  as  Fraunhofer's  lines.  Bring 
one  of  these  to  coincide  with  the  cross- 
hairs after  setting  the  prism  at  the  min- 
imum of  deviation,  and  read  the  vernier. 
It  will  be  found  that  the  minimum  for 
one  line  is  not  the  minimum  for  another. 
Measure  in  the  same  way  the  position 
of  several  of  the  more  prominent  lines  ; 
which  may  then  be  identified  by  the 
accompanying  table.  The  first  column 
gives  the  names,  the  second  their  wave- 
lengths, and  the  third  their  position  on 
the  map  of  Kirchhoff,  which  is  still  much 
used  as  a  standard.  The  line  A  is  about  the  extreme  limit  of  the 
red  end  of  the  spectrum,  and  can  only  be  seen  in  strong  sunlight. 


Name. 

W.L. 

K. 

A 

7605 

405 

a 

7185 

500 

B 

6867 

594 

C 

6562 

694 

a 

6276 

810 

D 

5892 

1005 

E 

5269 

1528 

h 

5183 

1634 

F 

4861 

2080 

G 

4307 

2855 

h 

4101 

3364 

HI 

3968 

3779 

SOLAR    SPECTROSCOPE.  153 

JB  is  therefore  often  mistaken  for  it.  C  is  a  sharply  marked,  but 
fine  line  in  the  red,  caused  by  hydrogen,  a  is  due  to  aqueous  vapor 
in  the  air,  and  is  most  conspicuous  about  sunset.  It  then  bears  a 
marked  resemblance  to  B.  J)  is  a  double  line  in  the  yellow,  due  to 
soda.  The  fine  lines  between  its  two  components  were  often  used 
as  tests,  until  it  was  shown  by  Prof.  Cooke  that  most  of  them  were 
due  to  aqueous  vapor.  E  is  a  close  double  line  in  the  green  in  the 
midst  of  a  group  of  double  lines,  some  of  them  very  close,  b 
consists  of  four  very  strongly  marked  lines,  three  of  them  due  to 
magnesium,  of  which  the  least  refrangible  is  b±.  They  contain 
several  fine  lines,  which  form  good  tests  of  the  power  of  a  spectro- 
scope. F  is  a  strong  line  in  the  blue,  like  (7,  due  to  hydrogen. 
G  lies  in  the  midst  of  a  group,  among  a  multitude  of  lines,  h  is 
fine  and  due  to  hydrogen,  and  H  consists  of  two  very  broad  lines, 
almost  at  the  limit  of  the  visible  spectrum. 

To  determine  the  indices  of  refraction  for  these  lines,  subtract 
from  the  reading  of  the  vernier  in  each  case  the  reading  when  the 
telescopes  were  opposite,  and  the  difference  D  gives  the  deviation. 
If  i  is  the  angle  of  incidence,  and  r  the  angle  of  refraction  for  the 
first  surface,  since  the  prism  is  at  the  minimum  of  deviation,  the 
angle  of  incidence  at  the  second  surface  will  equal  r,  and  the  an- 
gle of  refraction  i,  as  both  faces  are  equally  inclined  to  the  light. 
Again,  it  maybe  proved  by  Geometry,  that  calling  A  the  angle 
of  the  prism,  A  =  2r.  The  ray  is  deviated  at  each  surface  i  —  r, 
hence  the  total  deviation  D  =  '2(i  —  r)  =  2i  —  A,  or  i  = 
D).  If  the  index  of  refraction  equals  n^  sin  i  =  n  sin  r 


sin  \(A  4-  D) 

or  n  =   -  —  -.     .  j  -  -.     Compute  in  this  way  the  index  of  re- 
sin  -f)^cj, 

fraction  n  for  each  of  the  lines,  and  see  if  they  satisfy  the  theoret- 

B         C 
ical  formula  of  Cauchy,  n  =  A  -\  —  ^  -\  —  -±  =  A  -\-  Bx  -|-    Cx\ 

A  A 

calling  x  equal  to  the.  reciprocal  of  the  square  of  the  wave-length, 
and  A.,  B  and  (7,  constants  depending  on  the  particular  material 
of  which  the  prism  is  composed.  By  the  method  of  least  squares, 
p.  4,  the  most  probable  values  of  A,  B  and  C,  may  be  found,  and 
compared  with  observation  by  a  residual  curve.  To  insure  accu- 
racy, it  is  safer  to  remeasure  the  indices  again,  using  the  other  side 
of  the  graduated  circle  and  employing  the  mean.  By  using  a 


154  SOLAR    SPECTROSCOPE. 

hollow  prism  bounded  by  two  plates  of  glass,  the  indices  of  li- 
quids may  be  measured,  and  with  a  prism  of  quartz  the  relation 
of  the  ordinary  and  extraordinary  indices  to  the  wave-length,  es- 
tablished. 

To  obtain  really  valuable  results  in  this  experiment,  great  care 
is  necessary,  and  an  instrument  of  the  finest  construction.  The 
more  powerful  spectroscopes  contain  a  number  of  prisms,  thus 
giving  a  much  longer  spectrum.  In  some  the  light  passes  twice 
through  each  prism,  the  collimator  being  placed  immediately  over 
the  observing  telescope,  or  better,  united  with  it.  With  such  an 
instrument  a  vast  number  of  lines  may  be  seen  and  identified  by 
comparison  with  the  maps  of  Angstrom  or  Kirchhoff.  To  meas- 
ure the  exact  place  of  those  near  together,  it  is  better  to  determine 
accurately  the  position  of  two  or  three,  measure  the  rest  by  a 
spider-line  micrometer,  and  then  reduce  to  wave-lengths  by  inter- 
polation. 

The  distance  between  the  two  components  of  a  double  line  may 
also  be  readily  determined  by  the  same  instrument.  It  consists  of 
an  eye-piece,  in  which  are  two  vertical  spider  lines,  one  fixed,  the 
other  movable  by  a  micrometer  screw  the  number  of  whose  turns 
is  commonly  measured  by  notches  in  the  upper  part  of  the  field 
of  view,  and  the  fraction  of  a  turn  by  a  circle  divided  into  one 
hundred  parts,  attached  to  the  screw-head.  Both  wires  may  be 
moved  by  a  second  screw,  and  illuminated  by  a  light  placed  oppo- 
site a  piece  of  glass  inserted  in  one  side.  It  is  used  when  the  field 
is  dark,  to  render  the  lines  visible.  The  distance  between  two 
lines  may  be  measured  as  follows.  Call  the  screw  with  divided 
head,  A,  and  the  other,  B.  Bring  the  two  cross-hairs  to  coincide, 
and  read  the  micrometer-screw,  A,  repeat  several  times,  and 
take  the  mean.  Then  turn  B  until  the  fixed  hair  coincides  with 
one  line,  and  turn  B  until  the  movable  hair  coincides  with  the 
other.  The  reading  of  A  minus  that  previously  taken,  gives 
their  distance  apart.  After  setting  both  hairs,  their  position  is 
sometimes  reversed,  and  the  distance  through  which  A  has  been 
turned  equals  double  the  distance  between  the  lines.  It  is  a  good 
exercise  to  measure  all  the  lines  visible  in  a  small  portion  of  the 
spectrum,  and  then  compare  with  one  of  the  charts  mentioned 
above. 


LAW    OF    LENSES.  155 

If  the  sun  is  not  shining,  the  Bunsen  burner  may  be  employed 
instead,  using  the  platinum  wire  with  a  borax  bead  at  its  end. 
This  will  give  a  bright,  double  line,  coinciding  exactly  with  the 
dark  line,  D,  in  the  solar  spectrum.  Its  position,  and  the  interval 
between  its  components,  should  be  accurately  determined.  Spec- 
tra of  great  beauty  may  also  be  obtained  with  an  induction  coil 
by  allowing  the  spark  to  pass  between  terminals  of  different  met- 
als placed  in  front  of  the  slit.  Still  finer  effects  are  obtained  with 
the  electric  light. 

78.    LAW  OF  LENSES. 

Apparatus.  In  Fig.  60,  A  is  a  fishtail  burner,  attached  to  the 
end  of  a  bar  eleven  feet  long,  and  divided  into  tenths  of  an  inch. 
B  is  a  lens  of  two  feet  focus,  by  which  an  image  of  A  may  be 
projected  on  the  screen  O.  Both  B  and  C  are  movable,  and  carry 
pointers  to  show  their  distance  from  A. 

Experiment.  Place  C  at  the  end  of  the  bar,  and  B  just  100 
inches  from  A.  An  image  of  the  flame  will  be  formed  on  (7, 

which  is  then  moved  backwards  or 
•^0  A  &   forwards  until  the  exact  focus  is 

A T    found.     When   the  screen  is   too 

.,..,....^..,...,.-^.-..-. ..,....,.. .(..  »..J|  near,  it  will  be  noticed  that  owing 
to  chromatic  aberration,  the  edges 
of  the  image  are  red,  while  if  too 

distant  they  are  blue.  The  intermediate  position  may  thus  be 
found  with  great  accuracy.  Read  and  record  the  distance  A  C,  and 
repeat,  making  AB  successively  95,  90,  85,  etc.,  inches.  C  will 
approach  A  up  to  a  certain  point,  until  AB  —  J3C  equals  twice 
the  focal  length  of  the  lens.  Determine  this  point  more  exactly 
by  taking  a  number  of  readings,  moving  B  an  inch  at  a  time. 
Then  continue  to  diminish  AB  two  inches  at  a  time,  until  the 
image  falls  off  the  bar. 

Write  the  results  in  a  table  in  which  the  first  column  contains 
AB,  the  second  AC,  and  the  third  their  difference,  or  BC.  Now 
compute  the  true  value  of  BC  in  each  case,  and  insert  in  the 
fourth  column.  Calling  u  and  v  the  conjugate  foci,  or  AB  and 


156  MICROSCOPE. 

BC,  and  /the  principal  focus  of  the  lens,  -  -j  —  =  -  =  -j-=   -{- 
-          In  this  formula  AJBis  successively  made  equal  to  100,  95,  etc., 


inches,  and  /equals  one  fourth  the  minimum  value  of  AC.  By  a 
table  of  reciprocals  the  calculation  is  easily  made  by  subtracting 
the  reciprocal  of  AB  from  the  reciprocal  of/,  and  the  reciprocal 
of  the  difference  gives  BC.  Construct  two  curves  on  the  same 
sheet  of  paper  with  the  same  abscissas  AB,  a&d  ordinates  equal  to 
the  observed  and  computed  values  of  B  (7,  respectively,  and  their 
agreement  proves  the  correctness  of  the  formula. 

79.    MICROSCOPE. 

Apparatus.  The  importance  of  this  instrument  renders  it  de- 
sirable that  each  student  should  devote  considerable  time  to  its 
use.  For  this  reason,  in  a  large  laboratory  two  or  three  micro- 
scopes should  be  procured,  and  it  is  well  to  have  them  from  differ- 
ent makers,  so  that  the  student  may  be  accustomed  to  all  forms. 
For  example,  a  "Student's  Microscope,"  by  Tolles  or  Zentmayer, 
to  represent  the  American  instrument,  a  binocular  "  Popular 
Microscope,"  by  Beck,  for  the  English,  and  a  third  instrument  by 
Nachet  or  Hartnack,  for  the  Continental  form.  The  latter  is  very 
cheap  and  good,  but  not  having  the  Microscropical  Society's  screw, 
common  objectives  cannot  be  used  on  it  without  an  adapter.  It 
is  also  well,  if  it  can  be  afforded,  to  have  one  first-class  microscope- 
stand  for  work  of  a  higher  nature.  The  usual  appurtenances  de- 
scribed below  should  be  added,  but  need  not  be  duplicated,  also  a 
number  of  objectives  and  objects. 

The  following  description  will  serve  for  all  the  common  forms  of 
instrument.  A  brass  tube  or  body  is  attached  to  a  heavy  stand, 
so  that  it  can  be  set  at  an  angle,  or  moved  up  or  down.  In  its 
lower  end  the  objectives  are  screwed,  and  the  -lye-pieces  slide  into 
the  upper  end.  The  objectives  are  made  of  three  achromatic 
lenses,  by  which  a  short  focus  is  attained,  with  great  freedom  from 
aberration.  The  eye-piece  is  of  the  form  known  as  the  negative 
eye-piece,  and  consists  of  two  plano-convex  lenses,  with  their 
plane  surfaces  turned  upwards.  Below  is  placed  the  stage,  on 
which  the  object  is  laid  and  kept  in  place,  either  by  a  ledge,  or  by 
spring  clips.  In  the  larger  stands  the  object  may  be  moved  by 
two  racks  and  pinions  in  directions  at  right  angles,  or  revolved  by 
turning  the  stage.  It  is  very  desirable  that  this  rotation  should 
take  place  around  the  axis  of  the  instrument,  as  is  done  in  the  Eng- 
lish, but  not  in  the  American  instrument  mentioned  above.  Un- 
der the  stage  is  the  diaphragm,  a  plate  of  brass  with  a  number  of 


MICROSCOPE.  157 

circular  holes  in  it  of  different  sizes,  to  admit  light  more  or  less 
obliquely.  Below  it  is  a  mirror,  plane  on  one  side,  and  concave  on 
the  other,  by  which  light  may  be  reflected  upon  the  object. 

It  is  very  important  that  the  body  of  the  instrument  may  be 
raised  and  lowered  with  precision.  There  are  generally  two  ad- 
justments to  effect  this,  one  the  coarse  adjustment  to  move  it  rap- 
idly, which  is  commonly  a  rack  and  pinion,  or  a  simple  sliding 
motion  effected  by  hand,  and  a  fine  adjustment  which  is  used  for 
getting  the  exact  focus,  and  is  made  in  a  vaiiety  of  ways.  One  of 
the  best  is  by  a  movable  nose-piece,  or  the  lower  end  of  the  tube 
made  free  to  slide,  and  acted  on  by  a  lever,  which  may  be  moved 
by  a  screw.  In  a  second  form,  the  screw  acts  directly  on  a  part  of 
the  tube  itself,  and  sometimes  the  stage  is  raised  or  lowered.  If  the 
tube  is  moved,  it  should  be  raised  only  by  the  screw,  the  lowering 
being  effected  by  a  spring,  so  as  to  prevent  the  objective  from 
being  pressed  forcibly  against  the  object. 

To  show  the  use  of  each  instrument  used  in  connection  with 
the  microscope,  one  or  more  objects  should  be  selected  suited  to 
each,  and  numbered  as  in  the  foltowing  examples.  They  may 
then  be  distributed  among  the  various  microscopes,  according  to 
the  means  or  requirements  of  each  laboratory.  In  this  way  a 
student  acquires  a  better  knowledge  of  the  apparatus,  and  of  the 
proper  objects  to  which  each  appliance  is  best  suited,  than  he 
could  attain  in  weeks  of  unsystematic  work.  It  is  also  well  to 
examine  several  of  the  objects  described  below,  with  various 
methods  of  illumination,  to  learn  how  much  their  appearance  may 
be  thus  altered.  When  studying  a  new  object  it  should  always  be 
illuminated  in  various  ways,  and  viewed  first  with  a  low,  and  after- 
wards with  a  higher  power.  There  is,  however,  no  more  common 
mistake  than  to  suppose  that  objects  will  be  seen  better,  the  higher 
the  power.  On  account  of  the  difficulty  with  which  they  are 
used,  the  want  of  distinctness  and  of  sufficient  illumination,  the 
highest  magnifying  powers  must  be  reserved  for  special  occasions, 
and  the  lower  powers  commonly  employed,  especially  in  the  pre- 
liminary observation  of  common  objects.  To  save  the  eyes,  it  is 
better,  at  first,  not  to  use  a  microscope  very  long  at  a  time,  and  for 
the  same  reason,  they  should  both  be  kept  open.  If  possible, 
sometimes  one  eye  should  be  used,  and  sometimes  the  other. 

The  applications  of  the  microscope  have  been  so  extensive  that 
it  is  impossible  in  a  short  article  like  the  present,  to  give  more 
than  a  general  description  of  the  most  important.  The  student 
who  wishes  to  make,  a  specialty  of  this  instrument  is  therefore 
referred  at  once  to  some  of  the  works  devoted  especially  to  this 
subject.  For  instance,  the  treatises  of  Carpenter,  Hogg  and  Beale, 
particularly  the  work  by  the  latter  author,  entitled  "  How  to  Work 
with  the  Microscope."  The  same  remarks  apply  with  even  more 
force  to  Experiments  80  and  81. 


158  MICROSCOPE. 

Experiment.  1.  Ordinary  Method  of  using  the  Microscope. 
Set  the  microscope  in  an  inclined  position,  at  such  an  angle  that  it 
can  be  used  with  comfort.  The  tube  carries  an  objective  below, 
by  means  of  which  an  enlarged  image  of  the  object  is  formed,  and 
magnified  a  second  time  by  the  eye-piece.  Slip  into  the  upper 
end  of  the  tube  the  lowest  power  eye-piece,  that  is,  the  longest. 
The  objectives  are  contained  in  brass,  cylindrical  boxes,  with  screw 
covers.  They  must  be  handled  with  care,  as  they  are  very  expen- 
sive; the  higher  powers  consist  of  very  minute  lenses,  and  the 
glass  surfaces  must  never  be  touched,  lest  they  be  tarnished  or 
scratched;  the  lower  surface,  which  is  plane,  is  particularly  ex- 
posed to  injury.  Remove  the  cover  of  an  objective  whose  focus  is 
one  or  two  inches,  and  screw  it  into  the  tube.  Now  turn  the  mir- 
ror so  that  the  light  from  the  window  shall  be  reflected  along  the 
axis  of  the  instrument,  and  on.  looking  in,  a  bright  circle  of  light 
will  be  visible. 

Place  object  No.  1,  eye  of  a  fly,  on  the  stage,  and  raise  or  lower 
the  tube  until  it  is  distinctly  visible.  The  distance  between  the 
objective  and  object  should  be  somewhat  less  than  the  focal  length 
of  the  former.  Notice  that  the  eye  is  composed  of  a  multitude 
of  facets,  like  the  meshes  of  a  net,  each  one  containing  a  separate 
lens.  Sketch  some  of  them  in  your  note  book.  Try  the  other 
eye-piece,  which  will  give  a  somewhat  higher  power.  Then  re- 
move the  objective,  putting  it  back  in  its  box,  and  replace  it  by 
one  whose  focal  length  is  £  inch.  The  use  of  this  is  attended  with 
somewhat  greater  difficulty.  It  must  be  brought  very  near  the 
object,  but  not  in  contact,  or  it  would  very  likely  be  scratched,  or 
even  broken.  It  is  therefore  safest  to  bring  it  as  near  as  possible 
without  touching,  by  the  coarse  adjustment,  then  looking  through 
the  instrument  to  withdraw  it  until  the  distance  is  about  right,  and 
finally  focus  exactly,  by  the  fine  adjustment.  A  great  increase  of 
magnifying  power  is  thus  attained ;  add  to  the  description  and 
drawing  whatever  additional  is  visible.  Do  the  same  with  a  sec- 
ond object,  foot  of  the  Dytiscus. 

2.  Diaphragm.  Immediately  under  the  stage  is  a  brass  plate 
pierced  with  a  number  of  holes  of  different  sizes.  Its  object  is  to 
vary  the  amount  of  light  and  the  direction  in  which  it  comes. 
When  a  small  aperture  is  used,  all  the  light  comes  in  nearly  the 


MICROSCOPE.  159 

same  direction,  and  thus  renders  the  shadows  of  minute  objects 
more  distinct.  The  structure  of  delicate  objects  is  thus  some- 
times brought  out  very  beautifully,  where  a  large  aperture  con- 
ceals everything.  To  show  this,  try  object  No.  2,  proboscis  of  a 
horse-fly,  and  see  how  much  more  distinct  the  fine  hairs  at  the  end 
are,  with  the  small  aperture.  Also  the  diatom  Jsthmia  nervosa,  in 
which  the  markings,  although  perfectly  distinct  with  a  small 
aperture,  almost  disappear  when  the  diaphragm  is  turned  so  as  to 
admit  a  large  cone  of  light. 

3.  Oblique  Illumination.    Microscope  objectives  are  made  so 
that  they  will  transmit  rays  of  light  not  only  along  their  axis,  but 
also  when  falling  obliquely  on  them,  that  is,  they  will  receive  a 
cone,  the   angle  of  whose  vertex  is  called  the  angular  aperture  of 
the  objective.     For  the  higher  powers  this  angle  is  sometimes  very 
great,  170°,  175°,  or  even  177°.    With  them,  instead  of  placing 
the  mirror  immediately  underneath,  it  may  be  placed  on  one  side, 
and   the    object   illuminated   obliquely.     A   better  plan  is  by  an 
Amici's  prism  which  is  placed  below  the  object,  and  throws  the 
rays  obliquely  like  the  mirror.     The  advantage  in  this  case  is  like 
that  of  a  diaphragm,  only  greater,  shadows  being  strongly  cast,  and 
very  delicate  structure  rendered  visible.     This  effect  is  well  shown 
with  many  diatoms,  minute  siliceous  shells,  on  which  are  markings 
or  very  fine  parallel  lines,  used  frequently  as  tests.     Try  the  quarter 
inch   objective  on  specimens  No.  3,  Pleurosigma  formosum  and 
Pleurosigma  hippocampus.     First  use  direct  light  and   then  an 
oblique  illumination,  and  see  how  much  more  distinctly  the  mark- 
ings are  visible  in  the  second  case. 

4.  Opaque  Objects.     Some  objects,  especially  those  of  large  size, 
cannot  be  rendered  transparent,  and  sometimes  the  surface  only  of 
a  body  is  to  be  examined.     In  this  case  remove  the  mirror  from 
below  its  object,  and  place  it  above  on  a  stand,  turning  it  so  that 
the  light  shall  be  thrown  down  upon  the  object.    A  second  method 
is  to  use  for  the  same  purpose  a  large  lens  of  short  focus  called  a 
condenser.     If  the  observer  is  facing  the  window,  it  is  generally 
necessary  in  this  case,  to  place  the  object  nearly  horizontal,  in 
order  to  get  light  upon  it.     Try  both  these  methods  on  objects  No. 
4,  wing  of  a  butterfly,  and  section  of  bone  or  tooth,  viewing  the 
latter  also  by  transmitted  light. 


160  MICROSCOPE. 

5.  Lieberlvuhn.     Another  method  of  illuminating  opaque  ob- 
jects is  by  a  parabolic  mirror,  with  a  hole  in  its  centre,  through 
which  the  objective  is  passed.     This  device,  called  a  lieberkuhn,  is 
used  on  small  opaque  objects,  the  light  being  thrown  from  the 
mirror  below  upon  the  lieberkuhn,  and  by  it  reflected  upon  the 
object.     Try  specimen,  No.  4,  wing  of  butterfly,  thus  illuminated, 
also  some  common  objects,  as  a  bit  of  paper,  a  steel  scale,  etc. 

6.  ~Wenhairts  Parabolic  Condenser.     This  consists  of  a  block 
of  glass,  plane  below  and  parabolic  above.     It  is  placed,  instead 
of  the  diaphragm,  just  below  the  object,  which  is  at  its  focus,  so 
that  all  light  reflected  upon  it  by  the  mirror  below,  will  fall  on  the 
object  illuminating  it  obliquely.     The  central  rays  are  cut  off  by  a 
circle  of  metal  attached  to  the  condenser.     Objects  are  thus  shown 
bright  on  a  dark  background,  sometimes  producing  an  excellent 
effect,  though  generally  more   beautiful   than  useful.    See  No.  6, 
Arachnoidiscus  Ehreribergii. 

7.  Achromatic  Condenser.     The   mirror  below  the   object  is 
commonly  plane  on  one  side,  and  concave  on  the  other,  the  former 
reflecting  light  on  a  given  point  from  various  directions,  the  latter 
concentrating  that  received  from  a  single  point.     The  second  form 
is   more  commonly   used,  especially  with  artificial   light,  as   any 
point  may  thus  be  selected   as  the  source  of  illumination.     The 
same  effect  is  much  better  attained  by  placing  below  the  object  an 
objective  similar  to  that  above  it,  which  allows  only  those  rays 
parallel  to  its  axis  to  pass  through  both.     As  it  costs  too  much  to 
duplicate   all  the  objectives,  each  may  be  used  as  an  achromatic 
condenser  to  that  of  next  lowest  power.     This  is  a  very  favorite 
method  of  illumination,  especially  when  using  high  powers  on  dif- 
ficult objects.     Try  it  on  No.  7,  fragment  of  hair  and  Surirella 
gemma. 

8.  Polariscope.     One  other  method  of  illumination  remains  to 
be  described;  namely,  that  by  polarized  light.    To  use  this  to  the 
greatest  advantage,  Experiment  88  should  first  be  performed.    The 
light  is  polarized  by  a  Nicol's  prism,  placed  under  the  object  to  be 
examined  instead  of  the  diaphragm,  and  a  second  prism  or  analyzer 
is  placed  above  it,  either  slipping  it  over  the  eye-piece,  or  screwing 
it  onto,  and  just  above,  the  objective.     On  rotating  either  analyzer 
or  polarizer,  the  field  becomes  dark  when  their  planes  are  at  right 


MICROSCOPE. 


161 


angles,  nnd  brightest  when  they  are  parallel.  Sometimes  a  plate 
ofselenite  is  inserted  just  above  the  polarizer,  when  the  field  will 
assume  a  brilliant  color,  which  may  be  changed  into,  its  comple- 
mentary tint,  by  revolving  one  of  the  Nicol's  prisms.  To  examine 
any  object  by  polarized  light,  focus  it  as  usual,  and  turn  one  of  the 
prisms ;  the  most  marked  effect  is  generally  attained  when  they 
are  at  right  angles,  as  with  common  objects,  the  field  would  be 
perfectly  black,  while  any  doubly  refracting  medium  will  appear 
bright  in  places,  bringing  out  the  structure  with  great  beauty. 
Now  insert  the  plate  of  selenite,  and  the  uniform  tint  will,  in 
many  cases,  be  replaced  by  a  gorgeous  display  of  colors.  Examine 
the  following  objects,  No.  8,  section  of  cuttle-fish  bone,  quill,  sul- 
phate of  magnesia,  nitre  and  salicine.  In  fact,  any  crystalline  sub- 
stances not  in  the  monometric  system,  affect  polarized  light,  and 
the  same  may  be  said  of  many  organic  structures,  as  starch,  bone, 
hair  and  horn. 

9.  Binocular  Microscope.     To  avoid  the  fatigue  of  using  one 
eye  only,  and  to  obtain  the  stereoscopic  effect  due  to  two,  this 
instrument  is  employed.    A  small  prism  is  placed  just  above  the 
objective,  which   divides  the  light  into  two  portions,  which  pass 
along  two  tubes,  one  for  each  eye.     Two  eye-pieces  are  of  course 
used,  and  on  looking  through  them,  the  object  is  seen  as  in  the 
stereoscope,  standing  out  in  its  true  form.     The  distance  between 
the  eye-pieces  may  be  altered  by  lengthening  or  shortening  their 
tubes  by  a  pinion  acting  on  two  racks,  and  the  microscope  may  be 
rendered  monocular  by  merely  pushing  back  the  binocular  prism. 
This  instrument  is  best  suited  to  opaque  objects  not  requiring  a 
high  power.    Apply   it  to  objects  No.  9,  head  of  a  bee,  and  an 
injected  preparation  of  the  upper  portion 

of  the  lung  of  a  frog,  using  the  1  or  2  inch 
objectives  and  the  lieberkuhn. 

10.  Maltwood's  Finder.    Some  objects 
are   so   minute  that  they  are  found  only 
with  difficulty,  and  it  is   also  sometimes 
desirable  to  refer  back  to  a  certain  point 
of  an  object  and  reexamine  it.    A  Malt- 
wood's  finder  consists  of  a  photograph  on 

glass,  one  inch  square,  of  a  series  of  squares  numbered  as  in  Fig. 
11 


30 

31 

32 

51 

51 

51 

30 

31 

32 

52 

52 

52 

30 

31 

32 

52 

52 

52 

162 


MICROSCOPE. 


61,  all  the  lower  numbers  in  the  same  horizontal  row  being  the 
same,  also  all  the  upper  numbers  in  the  same  \ertical  column. 
In  other  words,  the  upper  numbers  give  the  abscissa,  the  lower 
the  ordinate,  of  each  square.  This  photograph  is  then  mounted 
on  a  slip  of  glass  like  any  other  object.  A  pin  or  stop  should  be 
attached  to  the  stage,  so  that  both  finder  and  object  may  al- 
ways be  placed  in  precisely  the  same  position.  Lay  the  object 
on  the  stage  and  bring  the  point  to  be  recorded  exactly  in  the 
centre  of  the  field,  using  a  moderately  low  power,  if  the  object 
is  not  too  minute.  Now  put  the  finder  in  the  place  of  the 
object,  taking  care  not  to  move  the  stage,  and  record  the  numbers 
of  the  square  in  the  centre  of  the  field.  If  at  any  time  the  same 
point  of  the  object  is  again  wanted,  the  finder  is  placed  on  the 
stage,  and  the  latter  moved  until  the  square  bearing  these  numbers 
is  again  in  the  centre  of  the  field.  Replacing  it  by  the  object,  the 
desired  point  should  be  at  once  visible.  Apply  this  method  to 
objects  No.  10.  First,  with  the  preparation  of  an  entire  insect,  as 
a  gnat,  record  the  numbers  corresponding  to  several  prominent 
points,  as  the  eye  and  the  end  of  the  proboscis.  Then  move  the 
stage  and  see  if  they  can  be  found  again.  Do  the  same  with  a 
slide  containing  a  single  minute  object  as  No.  7,  Surirella  gemma. 
Now,  placing  on  the  stage  an  object  containing  a  collection  of 
diatoms,  select  a  good  specimen,  sketch  its  position  and  that  of  the 
adjacent  ones,  and  see  if  it  can  be  found  again  from  its  numbers. 
The  numbers  corresponding  to  the  marked  points  of  some  of  these 
objects  should  be  recorded  on  a  card  placed  with  the  microscope, 
and  these  points  then  found  by  the  student. 

11.  Micrometers.  There  are  several  forms  of  this  instrument, 
which  is  used  for  measuring  minute  objects.  First,  the  stage-mi- 
crometer, which  consists  of  a  plate  of  glass  ruled  with  fine  lines  at 
equal  intervals  of  T^W  m-  °r  TOTT  mm'  The  lines  are  so  delicate 
that  it  is  often  difficult  to  find  them,  as  a  slight  difference  of  focus 
throws  them  out  of  sight.  Their  visibility  may  be  increased  by 
oblique  illumination,  or  by  using  a  small  aperture  in  the  diaphragm. 
The  eye-piece  micrometer  consists  of  a  coarser  scale  on  glass,  in- 
serted at  the  focus  of  the  eye-piece,  which,  in  the  negative  form,  lies 
between  the  lenses.  An  enlarged  image  of  it  is  thus  formed  in  the 
field,  which  is  used  like  an  ordinary  scale  to  measure  the  dimensions 


MICROSCOPE.  163 

of  objects.  To  reduce  them  to  fractions  of  a  millimetre,  lay  a  stage 
micrometer  on  the  stage  and  measure  its  divisions.  For  instance, 
if  7  hundredths  of  a  millimetre  equal  53.4  divisions  of  the  microm- 
eter, one  division  of  the  latter  will  equal  .00131  mm.  From  this 
any  measurement  made  with  the  micrometer  may  be  reduced  to 
millimetres.  If  the  microscope  has  a  draw-tube,  this  reduction  may 
be  simplified.  In  this  case,  the  tube  of  the  microscope  is  made 
double,  so  that  its  length  may  be  altered,  a  scale  showing  the 
extent  of  the  change.  The  power  is  increased  by  drawing  out  the 
tube,  and  the  divisions  of  the  stage-micrometer  being  enlarged, 
they  may  be  made  exactly  equal  to  any  desired  number  of  divi- 
sions of  the  eye-piece  micrometer.  Thus,  in  the  above  example, 
make  the  7  divisions  equal  to  56,  instead  of  53.4,  when  one  will  equal 
8,  or  each  division  of  the  eye-piece  micrometer  equal  one  eight 
hundredth  of  a  millimetre.  In  the  same  way  make  it  one  thou- 
sandth, and  altering  the  objective,  give  it  other  values,  recording 
in  each  case  the  reading  of  the  draw-tube. 

Now  make  a  number  of  measurements  of  objects  No.  11,  as 
directed  on  a  card,  which  should  accompany  the  specimens,  and 
reduce  them  to  millimetres.  Determine  also  the  thfckness  of  a 
hair,  of  a  filament  of  silk,  and  the  diameter  of  some  minute  holes 
made  in  a  piece  of  tin-foil  with  a  fine  cambric  needle.  In  the 
spider-line  micrometer,  the  two  hairs  are  brought  to  coincide  with 
the  ends  of  the  distance  to  be  measured,  and  the  interval  deter- 
mined, as  described  in  Experiment  77.  The  readings  should  be 
reduced  to  fractions  of  a  millimetre,  in  the  same  way  as  the  eye- 
piece micrometer.  Repeat  the  above  measurements,  and  see  if  the 
same  results  are  obtained  as  before.  The  magnitude  of  the  divi- 
sions of  both  the  spider-line  and  eye-piece  micrometers,  depends 
only  on  the  distance  from  the  objective  and  its  focus,  and  not  at 
all  on  the  eye-piece,  unless  a  negative  eye-piece  is  used,  and  the 
micrometer  inserted  between  the  field-  and  eye-lenses. 

12.  Goniometers.  A  spider-web,  or  filament  of  silk,  is  stretched 
in  the  eye-piece  across  the  field  of  view,  and  turned  so  as  to  coin- 
cide first  with  one  side,  and  then  with  the  other,  of  the  angle  to  be 
measured.  The  number  of  degrees  through  which  it  is  moved 
gives  the  required  angle.  A  graduated  circle  is  therefore  attached 
to  the  tube,  and  an  index  is  fastened  to  the  eye-piece.  Sometimes 


164  MICROSCOPE. 

the  object  is  moved  instead,  the  graduated  circle  being  attached  to 
the  stage.  To  test  this  instrument,  form  a  triangle  by  selecting 
three  points  on  any  object,  and  suppose  them  connected  by 
right  lines.  Measure  each  angle  two  or  three  times,  displacing  the 
object  and  eye-piece  after  each  measurement ;  take  the  mean,  and 
see  if  the  sum  of  the  three  equals  180°.  Now  measure  several 
angles  of  the  crystals,  No.  12,  chlorate  of  potash,  taking  care  that 
they  lie  flat,  otherwise  too  small  a  value  of  the  angle  will  be 
obtained. 

13.  Camera  Lucida.  It  is  often  important,  when  studying 
minute  objects  by  the  aid  of  the  microscope,  to  be  able  to  draw 
them  correctly.  For  this  purpose,  the  enlarged  image  must  be 
thrown  on  the  paper  in  such  a  way  that  both  may  be  distinctly 
seen  at  the  same  time.  This  is  done  most  simply  by  keeping  both 
eyes  open,  and  directing  one  towards  the  paper  the  other  through 
the  microscope,  when  the  image  and  paper,  may  be  brought  to- 
gether so  that  the  outlines  of  the  former  may  be  marked  on  the 
latter.  A  better  method  however,  is  by  the  camera  lucida,  which 
consists  of  a  minute  right-angled  prism  of  glass,  having  its  acute 
angles  equal  to  45°.  Place  the  microscope  horizontally,  adjust 
the  mirror  so  that  the  field  shall  be  bright,  and  apply  a  low  power 
to  object  No.  13,  a  flea.  Attach  the  camera  to  the  eye-piece  so 
that  on  looking  down  into  the  mirror,  a  reflection  of  the  object 
shall  be  thrown  on  a  sheet  of  paper,  placed  immediately  beneath, 
in  which  case  one  face  of  the  prism  will  be  horizontal  and  turned 
upwards,  the  other  vertical  and  turned  towards  the  microscope. 
The  prism  is  of  small  size,  so  that  it  will  cover  only  a  part  of  the 
pupil  of  the  eye,  and  bringing  the  latter  over  its  edge,  both  paper 
and  object  may  be  seen  simultaneously.  With  practice,  the  out- 
line may  be  marked  out  very  accurately,  but  at  first  it  is  diffi- 
cult to  see  the  pencil  at  the  same  time  as  the  object.  As  the  latter 
is  often  too  bright,  it  is  sometimes  better  to  incline  the  mirror  un- 
til the  field  is  darker.  Better  results  are  also  sometimes  obtained 
if  the  eye  is  raised  an  inch  or  so  above  the  prism. 

As  with  this  instrument,  the  brilliancy  of  the  object  generally 
renders  it  difficult  to  distinguish  the  pencil,  the  prism  is  sometimes 
replaced  by  a  small  piece  of  plate  glass,  which  reflects  the  image 
in  the  same  way  as  the  back  of  the  prism,  while  its  transparency 


MICROSCOPE.  165 

renders  the  pencil  visible  through  it.  The  latter  is  in  fact  gen- 
erally too  bright,  so  that  it  is  often  better  either  to  cast  a  shadow 
on  the  paper,  or  to  make  the  glass  of  a  neutral  tint,  to  render  it 
less  transparent.  Try  making  drawings  with  each  of  these  instru- 
ments, first  of  some  well  marked  object  as  the  flea,  and  afterwards 
of  some  fainter  object  as  the  anchor-like  spines  of  the  Synapta. 
Generally  only  the  outlines  should  be  drawn  with  the  camera,  and 
the  details  filled  in  by  the  eye.  After  drawing  the  object,  replace 
it  by  a  stage-micrometer,  and  draw  some  of  the  divisions  of  the 
latter,  which  thus  serve  as  a  scale,  by  which  the  magnitude  of 
different  parts  of  the  object  maybe  determined.  This  also  furnishes 
the  best  means  of  measuring  the  magnifying  power,  dividing  the 
dimensions  of  the  scale  as  drawn,  by  its  real  size.  This  will  be 
correct,  however,  only  when  the  distance  from  the  camera  to  the 
paper  is  just  10  inches.  In  other  cases  it  must  be  divided  by  the 
distance  in  inches,  and  multiplied  by  10,  to  reduce  it  to  the  stand- 
dard.  Make  this  measurement  with  the  1  inch  and  the  £  inch 
objectives,  and  two  of  the  eye-pieces,  also  if  there  is  a  draw-tube, 
make  the  magnifying  power  some  simple  number,  by  varying  the 
distance  between  the  objective  and  eye-piece. 

14.  Spectrum  Microscope.  This  consists  of  a  spectroscope  in- 
serted in  the  eye-piece  of  a  microscope  in  such  a  way,  that  the 
spectrum  of  very  minute  objects  may  be  obtained.  The  form  in 
most  common  use  is  that  devised  by  Sorby.  The  slit  replaces  the 
diaphragm,  and  is  partly  covered  by  a  right-angled  prism,  by  which 
a  second  spectrum,  from  a  light  at  one  side  of  the  eye-piece,  may  be 
compared  with  the  other.  The  prisms  are  placed  over  the  eye-lens, 
and  are  of  the  form  known  as  direct  vision,  in  which  the  deviation 
of  two  prisms  of  heavy  flint  glass  is  compensated  by  that  of  three 
crown-glass  prisms,  while  the  dispersion  is  only  partly  neutralized. 
Accordingly  a  spectrum  of  considerable  length  is  obtained,  while 
there  is  no  deviation  of  the  central  portion.  The  width  of  the 
slit  may  be  varied  by.a  screw,  and  its  length  by  a  sliding  stop.  An 
ingenious  scale  is  provided,  formed  of  two  Nicol's  prisms,  with  a 
plate  of  quartz  between  them,  and  placed  in  the  path  of  the  rays 
reflected  by  the  right-angled  prism.  They  absorb  from  the  visible 
spectrum  twelve  black  bands,  at  regular  intervals,  and  from  their 
position,  that  of  any  line  may  be  readily  determined.  To  use  this 


166  MICROSCOPE. 

instrument,  slip  it  on  the  end  of  the  microscope  in  the  place  of  the 
eye-piece,  and  place  object  No.  14,  human  blood,  on  the  stage  with 
a  one  or  two  inch  objective.  The  spectrum  will  now  be  seen  to  be 
traversed  by  two  marked  black  lines  in  the  red,  which  form  an 
excellent  test  for  the  presence  of  blood.  Their  position  may  be 
measured  with  the  scale,  by  attaching  the  latter  to  the  side  of  the 
eye-piece,  and  adjusting  the  prism  so  that  the  spectrum  for  one 
half  its  breadth  shall  be  traversed  by  strongly  marked  black  bands. 
Other  objects,  such  as  nitrate  of  didymium,  permanganate  of  pot- 
ash and  aniline  violet,  may  be  observed  in  a  similar  manner.  Care 
should  be  taken  to  make  all  the  light  pass  through  the  object, 
which  is  generally  best  accomplished  by  placing  a  cardboard 
diaphragm  with  a  small  hole  in  it,  on  the  stage  under  the  object. 
Liquids  are  placed  in  glass  tubes  or  cells,  which  may  be  closed 
hermetically. 

15.  Test- Objects.  The  principal  eiforts  of  microscope  makers 
are  now  directed  towards  the  objectives,  since  it  is  by  perfecting 
them  that  the  greatest  improvements  are  to  be  expected.  The 
best  method  of  judging  of  the  excellence  of  an  objective,  or  of 
comparing  those  of  different  makers,  is  by  trying  them  on  a  num- 
ber of  objects  called  test-objects,  some  parts  of  which  can  be  seen 
only  with  difficulty.  To  obtain  the  best  results  great  skill  is 
needed,  especially  in  arranging  the  illumination,  and  it  must  not 
be  forgotten  that  some  objectives  give  the  best  results  with  one 
class  of  objects,  others  with  another.  For  instance,  some  with  a 
large  angular  aperture,  give  fine  effects  with  objects  requiring  a 
very  oblique  illumination,  but  are  not  suited  to  those  of  considera- 
ble thickness,  requiring  great  depth  of  focus. 

When  an  objective  is  perfectly  corrected  for  chromatic  aberra- 
tion, and  a  plate  of  thin  glass  is  interposed  between  it  and  the 
object,  a  new  correction  for  color  becomes  necessary,  in  amount 
depending  on  the  thickness  of  the  glass.  This  is  commonly 
effected  by  varying  the  distance  of  the  front  lens  from  the  other 
two,  which  is  accomplished  by  turning  a  milled  head  near  the  end 
of  the  objective.  A  divided  circle  and  index  serve  to  mark  the 
position,  which  will  of  course  vary  with  each  different  object,  ac- 
cording to  the  thickness  of  the  covering  glass.  To  make  this 
correction,  adjust  the  objective  for  an  uncovered  object,  that  is, 


PREPARATION    OF    OBJECTS.  167 

set  the  index  at  zero  and  focus  it  on  the  object.  Then  turn  the 
milled  head  until  the  dust  on  the  upper  surface  of  the  covering 
glass  is  in  focus,  when  the  proper  correction  will  have  been  ap- 
plied. Focussing  again  on  the  object,  the  latter  will  be  more 
sharply  denned  than  before.  The  correction  for  covering  glass,  as 
it  is  called,  must  be  applied  to  all  objectives  of  higher  power  than 
i  inch,  to  get  the  best  effects,  especially  when  they  have  a  large 
angular  aperture.  Instead  of  moving  the  front  lens,  it  is  better  to 
have  it  fixed,  and  to  have  the  other  two  movable,  as  all  danger  of 
scratching  or  breaking  the  objective  and  object  by  bringing  them 
in  contact,  is  thus  avoided. 

Try  some  of  the  higher  power  objectives  with  the  test-objects 
No.  15.  One  of  the  most  common  tests  for  denning  power  is  the 
marking  of  the  scales  of  the  wood-flea  (Podura  plumbea),  which 
are  covered  wTith  delicate  epithelial  scales,  like  the  tiles  of  a  roof. 
Try  also  the  hair  of  the  Indian  bat,  and  of  the  larva  of  the  Der- 
mestes.  Some  of  the  Diatoms  described  above,  form  excellent 
test-objects.  The  valves  of  the  genus  Pleurosigma  are  covered 
with  fine  markings,  which  form  an  excellent  test  for  separating  or 
penetrating  power.  For  instance,  the  three  species,  formosum, 
hippocampus  and  angulatum,  form  a  series  of  increasing  difficulty, 
well  adapted  to  test  objectives  of  ordinary  power.  The  marking 
of  the  first  and  third  are  apparently  covered  by  three  series  of 
fine  parallel  lines,  dividing  the  surface  into  hexagons,  and  of  hippo- 
campus by  two  series,  forming  squares,  but  in  reality  probably  due 
to  a  multitude  of  very  minute  hemispheres  with  which  the  surface 
is  covered.  The  same  effect  may  be  seen  on  an  enlarged  scale,  in 
a  common  form  of  book-cover.  Probably  the  best  test  of  this 
kind  is  a  plate  of  glass  with  very  fine  lines  ruled  on  it.  M.  Nobert 
of  Griefs  wold  has  made  such  plates  with  a  series  of  bands  formed 
of  lines  at  various  intervals  up  to  a  112,000th  of  an  inch. 

80.     PREPARATION  OF  OBJECTS. 

Apparatus.  A  microscope  with  objectives  and  eye-pieces,  sev- 
eral vials  containing  the  substances  to  be  examined,  a  number  of 
glass  slips  three  inches  long  and  an  inch  wide,  some  of  which  have 
concave  centres,  that  is,  a  concavity  ground  out  on  one  side,  and 
some  circles  of  very  thin  glass. 


168  PREPARATION    OF    OBJECTS. 

Experiment.  To  examine  a  liquid  under  the  microscope,  dip  a 
glass  rod  or  tube  into  it,  and  place  a  small  drop  on  one  of  the  glass 
slides.  Cover  it  with  a  circle  of  very  thin  glass,  which  will  be 
held  in  place  by  capillarity,  and  wipe  off  the  superfluous  liquid 
carefully.  A  concavity  is  commonly  ground  in  the  centre  of  the 
slide  to  hold  more  liquid,  and  to  keep  the  cover  in  place.  Exam- 
ine the  following  objects  in  this  way,  describe  and  sketcli  them, 
and  compare  their  appearance  with  that  given  in  the  works  on  the 
Microscope,  referred  to  in  the  last  experiment.  A  drop  of  vinegar 
viewed  with  a  low  power,  is  seen  to  be  full  of  eels  in  active  motion. 
Milk  contains  multitudes  of  oil  globules,  which  when  united  form 
butter,  and  organic  matter  whose  appearance  furnishes  an  excellent 
test  of  its  purity.  Blood  is  a  curious  object  under  the  microscope. 
It  is  most  readily  obtained  from  the  finger  just  below  the  nail. 
With  a  quarter-inch  objective,  it  is  seen  to  consist  of  a  clear  liquid 
or  serum,  in  which  a  vast  number  of  blood-corpuscles  are  floating. 
These  are  circular  disks,  thicker  around  the  edge,  and  interspersed 
with  larger  white  globules.  In  its  natural  state  the  Jblood  is  too 
thick  to  be  conveniently  observed,  the  corpuscles  overlap,  and  soon 
begin  to  shrivel  up,  as  the  blood  dries.  If  diluted  with  water 
osmotic  action  ensues;  they  swell  up  and  sometimes  burst.  Salt 
water  is  therefore  preferable,  or  better  still,  the  serum  or  liquid 
portion  which  separates  from  the  clot  when  blood  coagulates. 

Powders  are  sometimes  viewed  dry,  but  generally  it  is  better  to 
wet  them,  as  they  are  thus  rendered  more  transparent.  Place  a 
very  minute  quantity  of  starch  on  a  slide,  add  a  drop  of  water, 
and  cover  with  a  piece  of  thin  glass.  Viewed  by  polarized  light, 
each  grain  is  seen  to  contain  a  black  cross,  which  changes  to  white 
on  rotating  the  analyzer.  This  cross  is  characteristic  of  starch^ 
and  often  serves  to  detect  its  presence.  It  is  best  seen  in  the  larger 
grains,  as  those  of  potato  starch,  and  assumes  brilliant  colors  if  a 
plate  of  selenite  is  interposed.  The  adulterations  of  coffee,  cocoa, 
etc.,  are  readily  detected  by  examining  them  in  powder  under  the 
microscope. 

It  is  often  necessary  to  pick  up  small  objects  under  water,  or  to 
capture  a  minute  animal  without  injuring  it.  A  good  example  of 
this  kind  is  the  little  Cyclops,  often  found  in  great  numbers  in 
common  pond  water,  especially  in  the  spring.  Collecting  the 


PREPARATION    OF    OBJECTS.  169 

water  in  a  white  porcelain  vessel,  as  a  large  evaporating  dish,  a 
close  examination  will  often  reveal  dozens  of  them.  Their  num- 
ber may  also  be  increased  by  filtering  a  considerable  quantity  of 
the  water  through  a  cloth,  which  retains  them,  and  from  which 
they  are  easily  washed  into  the  dish.  To  place  one  on  the  slide, 
take  a  small  glass  tube  about  half  a  foot  long,  close  one  end  by 
the  finger,  and  immerse  the  other  in  the  water.  Bring  it  near  the 
Cyclops  and  suddenly  remove  the  finger,  when  the  water  will  rush 
in,  carrying  the  animal  with  it.  Replacing  the  finger,  the  tube 
may  be  removed,  the  water  allowed  to  escape  a  drop  at  a  time, 
and  the  Cyclops  finally  deposited  on  the  slide.  Instead  of  a  slide 
with  concave  centre,  it  is  better  for  so  large  an  object  as  this,  to 
use  an  Animalcule-Cage.  This  consists  of  a  small  circle  of  glass, 
on  which  a  drop  of  water  containing  the  object  is  laid,  and  the 
cover  pressed  down  upon  it  by  means  of  a  brass  ring,  so  as  to 
leave  a  space  of  any  desired  degree  of  thickness.  Delicate  objects 
are  thus  protected  from  injury  by  crushing.  A  wonderful  variety 
of  animalcule  and  of  fungoid  growths  may  be  found  in  stagnant 
water,  or  sour  flour-paste,  in  fact  in  almost  any  decaying  animal  or 
vegetable  matter. 

Minute  air-bubbles  are  often  found  in  various  objects.  To  be- 
come familiar  with  their  appearance,  examine  a  drop  of  soap-suds, 
or  gum-water  containing  them.  They  look  like  black,  highly  pol- 
ished, metallic  balls,  with  a  broad,  dark  outline,  and  bright  centre. 

The  formation  of  crystals  is  readily  watched  under  the  micro- 
scope by  placing  a  drop  of  the  hot  saturated  solution  on  a  slide, 
and  allowing  it  to  cool.  Try  in  this  way  sugar,  phosphate  of 
soda,  and  oxalate  of  ammonia,  first  using  ordinary,  and  then  po- 
larized light. 

A  most  instructive  experiment  is  to  watch  the  circulation  of  the 
blood  in  the  foot  of  a  frog.  The  animal  is  first  rendered  insensible 
by  means  of  ether  or  chloroform,  then  put  in  a  linen  bag  and  well 
wet  with  water.  Draw  one  of  the  hind  legs  out  of  the  bag  and 
tie  it  down  upon  the  slide,  supporting  the  frog  on  a  piece  of  wood 
or  frog-plate.  Tie  threads  to  three  of  the  toes,  so  as  to  stretch  the 
membranes  between  them,  and  on  examining  it  with  a  half-inch 
objective  the  blood  corpuscles  will  readily  be  seen  passing  from 
the  arteries  through  the  capillaries  to  the  veins.  By  putting  alco- 


170  MOUNTING    OBJECTS. 

hol  or  salt  on  the  foot,  all  the  phenomena  of  inflammation  and  its 
cure  may  be  observed.  The  black  spots  distributed  over  the 
membrane  are  due  to  the  pigment.  The  circulation  may  also  be 
observed  in  the  tail  of  a  stickle-back,  or  other  small  fish,  or  of  a 
tadpole.  The  latter  animal,  when  very  small,  forms  a  beautiful 
object  with  a  low  power  and  binocular  microscope,  as  it  is  suffi- 
ciently transparent  to  render  visible  the  action  of  the  heart,  and 
other  internal  organs.  The  effect  is  also  improved  by  keeping  the 
tadpole  for  some  time  previously  without  food. 

Another  interesting  experiment  is  to  watch  the  ciliary  action, 
which  in  many  of  the  lower  animals  takes  the  place  of  the  circu- 
lation. Cilia  consist  of  minute  hairs,  which  vibrate  rapidly  back 
and  forth,  and  thus  establish  currents  in  the  liquid  in  contact  with 
them.  They  may  be  seen  by  scraping  a  little  of  the  mucus  from 
the  roof  of  the  mouth  of  a  frog,  or  better,  from  the  gills  of  an 
oyster  or  mussel. 

Most  solid  substances,  like  wood  or  bone,  are  best  seen  in  thin 
sections,  which  are  made  as  will  be  described  in  the  next  Experi- 
ment. Fine  filaments,  as  silk,  wool  or  hair,  are  viewed  by  trans- 
mitted light,  and  generally  give  better  effects  when  wet  with 
water  or  oil.  Some  solids,  especially  when  highly  colored,  are  best 
seen  as  opaque  objects,  with  a  condenser  or  lieberkuhn. 

81.     MOUNTING  OBJECTS. 

Apparatus.  Boxes  may  be  obtained  containing  all  the  appara- 
tus needed  for  mounting  objects,  such  as  glass  slips,  thin  glass 
covers,  Canada  balsam,  gold  size,  a  stand  on  which  slides  may  be 
heated,  a  whirling  table  for  making  cells,  section-cutting  appara- 
tus, and  other  devices  which  will  be  described  below. 

Experiment.  Objects  are  mounted  in  various  ways,  according 
to  their  size,  and  whether  they  are  best  seen  dry,  or  immersed  in 
some  liquid.  They  are  protected  by  a  circular  piece  of  glass, 
made  very  thin  on  account  of  the  short  working  distance  of  high- 
power  objectives.  These  circles  are  cut  from  a  sheet  of  thin  glass 
with  an  instrument  like  a  very  small  beam-compass,  the  point 
which  serves  as  a  centre,  being  replaced  by  a  flat  disk,  and  the 
pencil,  by  a  diamond.  Only  a  faint  scratch  is  needed,  but  some 
skill  is  required,  or  many  of  them  will  be  broken. 


MOUNTING    OBJECTS.  171 

Try  each  of  the  following  methods  of  mounting  objects,  and  if 
successful,  cover  the  slides  with  paper  and  label  them,  giving  also 
your  name  and  the  date.  Unless  the  object  is  very  thin,  or  if  it 
is  liable  to  be  injured  by  pressure,  it  must  be  protected  by  a  cell. 
This  consists  of  a  circular  or  square  enclosure,  on  which  the  thin 
glass  plate  is  laid,  so  as  to  leave  a  space  between  it  and  the  slide. 
Cells  may  be  made  of  various  materials,  as  paper,  cardboard,  or 
tinfoil,  and  fastened  to  the  glass  by  gum.  These  are  very  con- 
venient for  mounting  objects  dry,  especially  such  as  are  not  in- 
jured by  the  air.  Mount  in  this  way  some  crystals  of  bichromate 
of  potash.  Shallow  cells  may  be  made  of  Brunswick  black,  ap- 
plying it  with  a  brush.  They  are  best  made  in  a  circular  form  by 
Shadbolt's  apparatus,  in  which  the  slide  is  placed  on  a  small  turn- 
table, which  is  made  to  revolve  rapidly  by  drawing  the  forefinger 
of  the  left  hand  over  a  milled  head  attached  below,  while  the 
brush  is  held  in  the  right  hand.  If  the  plate  is  warmed,  the  black 
will  dry  rapidly,  and  the  thickness  of  the  cell  may  be  increased  by 
applying  successive  layers.  Make  several  such  cells  for  some  of 
the  objects  to  be  mounted  in  balsam,  as  described  below.  To 
preserve  a  liquid,  or  an  object  of  considerable  size,  thick  cells 
are  employed,  which  may  be  procured  ready-made  of  glass. 
They  may  be  cemented  to  the  slide  by  marine  glue,  warming  them 
sufficiently  to  melt  it,  removing  the  superfluous  glue  by  a  sharp 
knife,  and  washing  it  clean  with  a  solution  of  potash.  Fill  such  a 
cell  with  some  liquid,  as  vinegar,  and  fasten  on  the  cover  with 
marine  glue.  Take  great  care  thatno  air  bubbles  enter,  and  that 
the  joints  are  perfectly  tight. 

The  best  method  of  mounting  the  parts  of  insects,  sections  of 
wood  or  bone,  and  in  fact  most  substances,  is  in  Canada  balsam. 
The  object,  as  the  foot  of  a  fly,  must  first  be  dried  and  freed  from 
air-bubbles.  For  this  purpose  it  should  be  heated  nearly  to  the 
boiling  point  of  water,  or  placed  under  a  bell-glass  containing 
concentrated  sulphuric  acid.  To  remove  the  air  it  should  be 
soaked  in  turpentine  and  gently  warmed ;  a  much  more  effective 
method  is  to  place  the  whole  under  the  receiver  of  an  air-pump 
and  exhaust.  Now  lay  the  slip  of  glass  on  a  little  stand  of  brass, 
and  heat  it  by  means  of  a  spirit-lamp,  or  Bunsen  burner.  Take  up 
a  little  Canada  balsam  on  the  end  of  an  iron  wire,  and  lay  it  on 


172  MOUNTING   OBJECTS. 

the  slide,  when  the  heat  will  render  it  perfectly  fluid.  Pick  up  the 
object  on  the  point  of  a  needle,  immerse  it  in  the  balsam,  and  then 
cover  it  with  a  piece  of  thin  glass.  Great  care  must  be  taken  that 
both  slide  and  covering-glass  are  perfectly  clean,  and  that  no  dust 
gets  into  the  balsam,  as  otherwise  the  object  will  be  much  dis- 
figured when  viewed  under  the  microscope.  The  main  difficulty 
is  to  prevent  air-bubbles  remaining  on  the  slide.  If  present,  they 
may  be  removed  by  a  cold  wire,  or  burst  by  touching  them  with  a 
hot  needle.  The  covering-glass  must  be  lowered  into  place  very 
slowly,  or  bubbles  will  adhere  to  its  surface.  The  whole  is  then 
put  away  to  harden  under  pressure,  and  the  superfluous  balsam 
afterwards  removed  by  the  aid  of  a  little  turpentine. 

The  structure  of  objects  of  large  size  is  generally  best  seen  by 
cutting  thin  sections  of  them,  so  that  they  may  be  rendered  nearly 
transparent,  and  be  viewed  by  transmitted  light.  Soft  substances, 
as  vegetable  or  animal  tissues,  may  be  cut  with  a  sharp  knife  or 
scissors,  or  better,  with  a  Valentin's  knife,  which  has  two  parallel 
blades  whose  distance  apart  may  be  varied  by  a  screw.  They 
should  be  well  wet  with  water  or  glycerine,  or  the  section  will 
adhere  to  them. 

Harder  substances,  as  wood  or  horn,  are  cut  in  thin  sections  by 
forcing  them  through  a  hole  in  a  thick  brass  plate,  cutting  off  the 
projecting  portion,  pushing  it  through  a  little  farther,  and  cutting 
again.  By  means  of  a  screw,  sections  of  any  desired  thickness 
may  thus  be  obtained.  Cut  longitudinal  and  transverse  sections 
of  a  piece  of  pine  wood,  first  soaking  it  in  water,  and  mount  them 
in  Canada  balsam.  Cut  also  some  thin  transverse  sections  of 
hair  by  fastening  a  number  of  them  together  with  gum  so  as  to 
form  a  solid  mass ;  cut  a  thin  section,  and  then  dissolve  the  gum 
in  water. 

To  cut  a  thin  section  of  still  harder  substances,  as  bone,  quite  a 
different  method  must  be  employed.  A  thin  piece  is  first  cut  off 
with  a  fine  saw,  such  as  is  used*  for  cutting  metals ;  it  is  then  filed 
thinner,  and  finally  ground  down  to  the  required  thickness  with 
water  between  two  hones.  On  examining  the  section  thus  obtained, 
it  will  be  found  covered  with  scratches,  which  must  be  removed 
by  grinding  it  on  a  dry  hone,  and  afterwards  polishing  it  -on  a  sheet 
of  plate  glass.  Prepare  two  such  sections,  soak  one  in  turpentine 


FOCI    AND    APERTURE    OF    OBJECTIVES.  173 

and  remove  the  air  by  means  of  a  pump,  and  then  mount  both  in 
Canada  balsam.  The  difference  in  their  appearance  will  be  very 
marked,  the  one  from  which  the  air  has  not  been  removed  appear- 
ing full  of  black  spots  or  lacunaB,  formerly  called  bone  corpuscles. 
They  are  really  cavities  filled  with  air,  which  in  the  second  speci- 
men is  replaced  by  the  turpentine. 

This  experiment  is  well  supplemented  by  performing  some 
dissections  of  animal  and  vegetable  substances,  injecting  tissues, 
and  mounting  thin  sections  of  them. 

82.     Foci  AND  APERTURE  OF  OBJECTIVES. 

Apparatus.  Two  instruments  are  needed  for  this  Expeiiment. 
First,  a  microscope  with  a  positive  eye-piece,  a  spider-line  or  eye- 
piece-micrometer, and  a  stage-micrometer,  also  several  objectives 
to  be  measured.  To  measure  the  angular  aperture,  a  graduated 
circle  is  employed  with  an  arm  and  index,  to  which  is  attached  a 
short  brass  tube,  like  the  body  of  a  microscope.  The  objective  to 
be  tested  is  screwed  into  one  end  of  this  tube,  and  a  positive  eye- 
piece slipped  into  the  other.  The  tube  is  made  so  short  that  when 
the  objective  is  directed  towards  a  distant  object,  the  image  formed 
may  be  viewed  by  the  eye-piece.  To  obtain  a  higher  magnifying 
power,  the  eye-piece  may  be  replaced  by  a  compound  microscope, 
like  that  used  in  Experiment  No.  20.  To  obtain  an  accurate 
measurement  when  the  object  observed  is  not  very  distant,  it  is 
essential  that  the  end  of  the  objective  should  lie  in  the  axis  of  the 
circle.  This  is  most  readily  accomplished  by  means  of  a  ledge,  on 
which  a  vertical  plate  of  glass  may  be  placed  with  its  front  face 
over  the  axis  of  the  circle.  The  objective  is  then  brought  up  in 
contact  with  it,  the  tube  clamped,  and  the  glass  removed. 

Experiment.  To  measure  the  focal  length  of  an  objective  it  is 
assumed  that  two  of  the  laws  of  simple  lenses  hold  for  a  com- 
pound lens.  First,  that  the  sum  of  the  reciprocals  of  the  conju- 
gate foci  equals  the  reciprocal  of  the  principal  focus,  and  secondly, 
that  the  ratio  of  the  magnitudes  of  the  object  and  image  equals  the 
ratio  of  the  conjugate  foci.  This  assumption  is  not  strictly  cor- 
rect, and  valuable  work  might  be  done  in  determining  the  amount 
of  the  deviation.  Screw  the  objective  to  be  measured  upon  the 
microscope,  and  measure  the  divisions  of  the  stage-micrometer, 
with  the  spider-line  micrometer.  Reduce  to  absolute  measure- 


174  FOCI    AND    APERTURE     OF    OBJECTIVES. 

ments  from  the  magnitude  of  the  parts  of  the  micrometers,  or  if 
these  are  not  given,  determine  them  from,  the  Dividing  Engine, 
Experiment  No.  21.  This  reduction  may  be  avoided  by  using  two 
similar  eye-piece  micrometers,  A  and  B.  Measure  several  divi- 
sions of  A  with  J?,  and  call  the  mean  of  the  readings  m.  Meas- 
ure, in  the  same  way,  B  with  A,  and  call' the  mean  reading  mf. 
The  true  reading,  w,  will  be  the  mean  proportional  of  these  two. 
Of  course  if  the  micrometers  are  precisely  alike,  m  will  equal  mf. 
Now  measure  the  distance  between  them,  and  call  the  distance  D. 
Then  if/  equals  the  focal  length  of  the  objective,  and/',/"  its 

111  f 

conjugate  foci,/'  +/"  =  Z>,  jr  +  jr,  =  j,  and   V,  =  n.  From 

which  /—  D,    _|_  j>2,  and  knowing  D  and  ?i,/may  be  deduced. 

The  number  given  by  the  maker  is  generally  greater  than  the  true 
focal  length  of  the  objective,  and  this  experiment  affords  an  excel- 
lent means  of  correcting  it.  To  show  the  value  of  such  measure- 
ments, and  the  accuracy  attainable  by  them,  see  an  article  by  Prof. 
Cross,  Journ.  Frank.  Inst.,  Vol.  LIX.,  p.  401.  Useful  work  might 
also  be  done  by  varying  Z>,  and  noting  the  effect  on  /  also  by 
changing  the  correction  for  cover,  or  distance  between  the  lenses. 
To  measure  the  angular  aperture  of  an  objective,  screw  it  into 
the  end  of  the  tube  attached  to  the  graduated  circle,  set  a  plate  of 
glass  on  the  ledge,  and  bring  the  objective  against  it.  The  front 
surface  of  the  lens  will  then  be  just  over  the  axis  of  the  circle. 
Now  clamp  the  tube,  remove  the  plate  of  glass,  and  slide  the  eye- 
piece or  small  compound  microscope  into  place.  Bringing  it  near 
the  objective,  an  image  of  outside  objects  is  seen,  the  whole  in 
fact,  forming  a  telescope  with  the  objective  for  an  object-glass. 
The  field  of  view  is  seen  to  be  bounded  by  a  circle  whose  true 
angular  diameter  gives  the  aperture  of  the  objective.  Select  some 
strongly  marked  vertical  line,  as  the  sash  of  the  window,  and  notice 
that  as  the  objective  is  turned  from  side  to  side,  the  image  of  this 
line  moves  also.  Bring  it  to  coincide  first  with  one  edge  of  the 
circle,  and  then  with  the  other.  The  difference  in  the  reading  of 
the  index  in  the  two  cases  equals  the  angular  aperture.  Repeat 
this  measurement  with  several  other  objects. 


TESTING  PLANE  SURFACES.  175 

83.    TESTING  PLANE  SURFACES. 

Apparatus.  A  stand  carrying  two  telescopes,  which  may  be 
placed  opposite  each  other,  or  set  at  right  angles.  The  eye-piece 
of  one,  which  acts  as  a  collimator,  is  replaced  by  a  plate  of  brass 
pierced  with  a  very  fine  hole.  This  is  placed  exactly  at  the  focus 
of  the  object-glass,  and  being  illuminated  by  a  lamp,  forms  a  blight 
point  of  light  or  artificial  star.  The  optical  circle  might  be  used 
for  this  experiment,  but  the  graduated  circle  is  not  needed,  and  it 
is  better  to  have  telescopes  of  larger  size.  A  millimetre  scale  is 
also  wanted,  a  prism,  a  sextant-glass,  a  piece  of  plate  or  window 
glass,  and  a  lens  of  very  long  focus. 

Experiment.    Make  the  same  adjustment  for  parallel  rays  as  is 
described  in  Experiment  72.     That  is,  focus  the  observing  telescope 
carefully  on  some  distant  object  as  a  star,  and  turn  it  toward  the 
collimator.     An  image  of  the  hole  or  artificial  star  at  the  further 
end  of  the  latter  will  now  be  visible,  but  it  will  generally  be  out 
of  focus.    Draw  it  towards,  or  from  its  object-glass  until  accurately 
focussed,  when  it  should  appear  as  a  very  minute  circle  of  light, 
like  a  star.     Measure  with  the  millimetre  scale  the  distance  be- 
tween two  points,  one  on  the  eye-piece,  the  other  on  the  end  of 
the  tube  in  which  it  slides.     Throw  the  star  out  of  focus  by  mov- 
ing the  eye-piece,  and  focus  again  ;  repeat  ten  times,  and  take  the 
mean  of  the  distances  between  the  two  points.     Now  set  the  teles- 
copes at  right  angles,  and  place  the  surface  to  be  tested  at  the 
intersection  of  their  axes,  equally  inclined  to  each,  and  vertical. 
The  image  of  the  star  reflected  in  the  surface,  will  then  fall  in  the 
centre  of  the  field,  and  if  the  surface  is  perfectly  plane  will  be  as 
distinct    as  that  previously  obtained,  although   fainter.     In  gen- 
eral, however,  it  will  be  a  little   out  of  focus,  due  to  the  curva- 
ture of  the  surface.     In  this  case  move  the  eye-piece,  focus  ten 
times  as  before,  and  take  the  mean  reading  of  the  distance  between 
the  two  marked  points.     Measure  the  focal  length  of  the  observ- 
ing telescope,  or  the  distance  from  its  object-glass  to  the  cross-hairs, 
also  the  angle  between  the  axes  of  the  collimator  and  observing 
telescope,  unless  this  is  fixed  at  90°.     Call  F  the  focal  length,  d 
the  change  in  position  of  the  eye-piece,  or  difference  of  the  means 
of  the  two  sets  of  ten  observations,  v  the  angle  between  the  axes, 
a  the  distance  from  the  objective  of  the  telescope  to  the  plane  sur- 


176  TESTING    PLANE    SURFACES. 

face,   and   R  the  required    radius   of   curvature.      Then   R    = 

2  • 7— - — ; — ,  or  as  d  is  generally  very  small  compared  with 

a  cos  MJ  j        j 

F* 

F,  it  is  often  sufficiently  accurate,  if  v  =  90°,  to  take  R  =  2.83— j— 

If  the  surface  is  concave,  the  eye-piece  has  to  be  pushed  in,  if  con- 
vex, out.  Test  in  the  same  way  the  other  plane  surfaces,  also  the 
two  sides  of  the  lens.  Any  distortion  of  the  image  is  due  to 
irregularity  of  the  surfaces,  as  is  well  shown  by  trying  a  piece  of 
window  glass. 

The  parallelism  of  two  plane  surfaces,  like  those  of  the  sextant- 
glass,  is  well  tested  in  the  same  way.  If  both  surfaces  are  per- 
fectly plane  and  parallel,  only  a  single  image  is  formed,  otherwise 
there  are  two,  one  from  each  face.  The  angle  A,  between  them, 
may  be  determined  from  the  divergence  of  the  images,  by  the 

formula  A  —  „ —      — ,  in  which  n  is  the  index  of  refraction,  D 

~J(  COS  ^* 

the  angle  between  the  images,  and  r  the  angle  of  refraction  of 
the  light  in  the  glass.  The  latter  is  known  from  the  equation 
sin  ^v  =  n  sin  r,  in  which  v  is  the  angle  between  the  telescopes. 
If  v  =  90°,  A  =  .267  D,  and  if  v  =  0°,  A  =  .33  D.  If  the  sur- 
faces are  curved,  it  is  also  possible  to  determine  the  curvature  of 
both,  from  the  two  images,  but  the  problem  is  then  much  more 
complex. 

Another  method  is  to  place  the  telescopes  opposite  each  other, 
and  cover  half  their  object-glasses  with  the  plate  to  be  tested.  If 
the  two  surfaces  are  plane  and  parallel,  no  effect  will  be  produced. 
If  they  are  inclined,  they  form  a  prism,  and  cause  a  second  image. 
If  JO  is  the  angular  interval  between  the  two  images,  and  A  the 

angle  between  the  two  faces,  A  =  n  ^  or  if  n  =  1.5,  A  = 

2Z>.  Comparing  this  formula  with  that  given  above,  it  is  evident 
this  method  possesses  only  about  one-seventh  the  delicacy  of  the 
other,  since  for  a  given  value  of  A,  the  divergence  of  the  two 
images  is  only  a  seventh  part  as  great.  The  delicacy  of  the 
method  by  reflection  may  also  be  increased  indefinitely  by  increas- 
ing v.  If  the  surfaces  are  curved  they  act  like  a  lens,  and  throw 
the  image  out  of  focus.  The  problem  now  becomes  indeterminate, 


TESTING    PLANE    SURFACES.  177 

as  there  is  only  one  equation  to  determine  the  curvature  of  both 
faces.  The  focus  of  the  equivalent  lens  may,  however,  be  found 
by  measuring,  as  before,  the  change  in  position  of  the  eye-piece, 

when  the  focus  /  will  equal  -         -g —     — -.     If  d  is  very  small, 

F* 
f  =  ~jfi  which  is  the  best  method  of  measuring  the  focus  of  a 

very  flat  lens.  Thus,  if  F  =  24  inches,  and  d  =  one  inch,/ will 
equal  nearly  fifty  feet.  As  in  the  case  of  reflection,  any  irregular- 
ity of  the  surfaces  produces  a  distortion  of  the  image.  Test  in 
this  way  the  plates  of  glass,  and  the  lens. 

Still  another  method  of  testing  the  flatness  of  a  glass  plate  is  to 
form  Newton's  rings,  using  the  monochromatic  light  of  a  soda 
flame.  Very  slight  irregularities  in  the  surface  will  then  appear 
covered  with  yellow  and  black  rings,  like  contour  lines. 

As  it  is  often  desirable  to  increase  the  reflecting  power  of  a 
plane  surface  of  glass  when  it  is  to  be  used  as  a  mirror,  the  most 
common  methods  of  silvering  are  here  appended.  A  looking-glass 
is  made  by  covering  the  back  of  the  glass  with  an  amalgam  of 
mercury  and  tin,  as  follows.  Lay  a  sheet  of  tinfoil  the  size  of 
the  glass  to  be  silvered  on  a  level  surface,  and  pour  some  mercury 
upon  it,  making  it  spread  over  the  whole  surface  with  a  hare's  foot. 
Lay  a  sheet  of  paper  on  it,  and  the  glass  over  all.  Then  draw  the 
paper  slowly  out,  when  the  mercury,  as  it  is  exposed,  will  unite  with 
the  glass,  and  the  paper  will  remove  any  adhering  dust.  Special 
care  is  needed  that  the  tin,  mercury,  paper  and  glass,  should  be 
perfectly  clean,  and  that  no  bubbles  remain  under  the  glass.  Some- 
times the  paper  is  dispensed  with,  and  the  glass  slid  on  over  the 
mercury,  bringing  it  first  in  contact  at  one  corner.  It  is  then  sub- 
jected to  pressure,  and  set  up  on  edge  to  drain.  It  is  best  to 
keep  this  mercury  by  itself,  as  if  used  for  other  purposes,  it  is 
difficult  to  remove  the  tin,  which  gives  much  trouble  by  adhering 
to  any  glass  surfaces  with  which  it  is  brought  in  contact.  When 
a  bright  light  is  viewed  in  such  a  mirror,  holding  it  very  obliquely, 
a  large  number  of  images  is  seen.  The  first,  reflected  from  the 
front  surface  is  faint,  the  second  from  the  mercury  is  strongly 
marked,  and  these  are  succeeded  by  many  others,  caused  by  suc- 
cessive reflections,  and  growing  fainter  and  fainter  until  they  fi- 
nally become  invisible. 
12 


178  TESTING    TELESCOPES. 

To  obtain  a  single  image  only,  sometimes  a  plate  of  black  glass 
is  used,  or  the  lower  surface  is  covered  with  black  paint,  or  better, 
since  much  light  is  lost  in  this  way,  the  front  surface  is  covered 
with  a  deposit  of  metallic  silver.  One  method  of  doing  this  is  by 
dissolving  10  grms.  of  pure  nitrate  of  silver  in  20  grins,  of  water, 
and  adding  5  grms.  of  ammonia.  Filter,  add  35  grms.  of  alcohol  of 
specific  gravity  .842,  and  10  drops  of  oil  of  cassia.  Cover  the  plate 
with  this  mixture  to  a  depth  of  quarter  or  half  an  inch,  and  add  6 
to  12  drops  at  a  time  of  a  mixture  of  1  part  of  oil  of  cloves  to  3  of 
alcohol,  until  the  whole  surface  is  covered  with  the  precipitated 
silver.  The  plate  is  then  dried,  cleaned  and  polished.  Various 
other  receipts  are  recommended,  some  using  starch,  sugar,  or  tar- 
taric  acid,  instead  of  oil  of  cloves  to  precipitate  the  silver.  Proba- 
bly much  more  depends  on  the  practice  and  skill  of  the  experi- 
menter than  on  the  details  of  the  different  formulas.  Liebig  em- 
ploys a  liquid  formed  by  adding  soda-ley  of  sp.  gr.  1.035  to  45  cm.8 
of  an  ammoniacal  solution  of  fused  nitrate  of  silver,  and  dissolving 
the  precipitate  formed  by  adding  ammonia  until  the  volume  equals 
145  cm.8.  Add  5  cm.8  of  water,  and  shortly  before  using  it,  mix 
with  one  sixth  to  one  eighth  of  its  volume  of  a  solution  of  sugar  of 
milk,  containing  1  part  of  sugar,  to  10  of  water.  Flood  the  glass 
to  a  depth  of  half  an  inch,  and  it  will  soon  become  covered  with 
a  thick  coating  of  silver. 

Another  method  of  making  reflectors,  is  by  platinizing  glass,  or 
covering  it  with  a  layer  of  metallic  platinum.  This  is  accom- 
plished by  covering  the  surface  with  chloride  of  platinum  with  a 
brush,  reducing  it  to  a  metallic  state  by  oil  of  lavender,  and  heat- 
ing it  in  a  muffle. 

84.    TESTING  TELESCOPES. 

Apparatus.  A  long  darkened  chamber  with  a  small  aperture 
at  the  farther  end,  through  which  the  light  of  the  sky,  or  of  a  lamp, 
shines.  A  long  empty  water-pipe,  or  unused  flue,  answers  very 
well  for  this  purpose,  but  if  this  cannot  be  obtained,  a  large  black 
screen  with  a  small  hole  in  it  may  be  placed  at  the  farther  end  of 
the  room,  and  a  short  tube  blackened  on  the  interior,  used  to  cut 
off  the  stray  light.  A  double  length  may  be  obtained  by  placing 
a  plane  mirror  at  the  farther  end  of  the  room,  and  the  screen  close 
to  the  observer.  A  telescope  to  be  tested,  which  should  have  an 


TESTING     TELESCOPES.  179 

object-glass  at  least  three  inches  in  diameter,  is  also  needed.  It  is 
composed  of  two  lenses,  one  concave,  of  flint,  the  other  convex,  of 
crown  glass.  The  focus  of  the  latter  will  be  about  three-fifths 
that  of  the  two  together.  Suppose  this  is  three  feet,  then  the 
focal  length  of  the  crown  glass  will  be  about  22  inches.  Procure 
two  similar  lenses  of  20  and  24  inches  focus,  respectively.  Com- 
bining the  first  with  the  flint  gives  an  under-corrected,  while  the 
other  gives  an  over-corrected  lens. 

Experiment.  The  principal  defects  to  be  sought  for,  are  striae 
or  irregularity  of  the  glass,  spherical  aberration  or  incorrect  form, 
chromatic  aberration  or  imperfect  correction  for  color,  imperfect 
annealing  of  the  lenses,  and  wrong  centering  or  want  of  coinci- 
dence of  their  axes  with  that  of  the  telescope. 

First,  to  test  for  striae,  direct  the  telescope  towards  the  artificial 
star  or  minute  point  of  light  at  the  farther  end  of  the  room.  Then 
remove  the  eye-piece,  and  placing  the  eye  in  the  axis  of  the  in- 
strument a  bright  circle  of  light  will  be  seen,  which  will  cover  the 
whole  object-glass  when  the  eye  is  exactly  at  the  principal  focus. 
If  now  the  glass  is  free  from  veins,  striae,  or  other  imperfections, 
this  circle  will  appear  perfectly  uniform,  otherwise  the  striae  are 
shown  in  a  very  marked  manner.  To  determine  whether  they  are 
caused  by  the  crown  or  flint  lens,  remove  the  latter,  and  see  if 
they  still  remain.  Test  in  the  same  way  the  other  two  convex 
lenses,  and  sketch  any  striae  that  may  be  present.  Some  cheap 
cosmorama  lenses  are  made  of  common  plate-glass,  in  which  case 
they  are  often  full  of  parallel  striae,  running  in  the  direction  in 
which  the  glass  was  rolled. 

To  test  for  spherical  aberration,  place  the  eye  a  little  beyond 
the  focus,  and  pass  a  card  through  the  latter.  Since  all  the  rays 
would  intersect  at  the  focus  if  there  were  no  spherical  aberration, 
the  light  would  be  instantly  extinguished  when  the  card  covered 
this  point.  In  practice,  however,  the  bright  circle  of  light  assumes 
the  appearance  of  a  curiously  shaped  surface  of  revolution,  from 
which  the  form  of  the  lens  is  readily  determined. 

To  test  for  chromatic  aberration,  examine  the  image  of  the 
artificial  star  with  an  eye-piece,  precisely  as  when  looking  through 
the  telescope  at  a  real  star.  If  the  lens  was  perfectly  achromatic, 
a  very  minute  circle  of  light  would  be  obtained,  which  would 
enlarge  on  pushing  the  eye-piece  in  or  out,  remaining  all  the  time 


180  TESTING     TELESCOPES. 

perfectly  colorless.  The  change  in  size  is  due  to  the  fact,  that  the 
rays  of  light  form  a  cone  of  which  the  object-glass  is  the  base,  and 
the  focus  the  apex.  The  field  of  view  is  really  a  section  of  this 
cone  at  right  angles  to  its  axis.  If  an  uncorrected  lens  is  used, 
since  the  violet  rays  are  more  bent  than  the  red,  they  form  a  cone 
with  vertex  nearer  the  object-glass;  accordingly,  if  the  eye-piece  is 
pushed  in,  the  centre  of  the  circle  will  be  violet,  and  the  exterior 
red.  Owing  to  the  unequal  dispersion  of  different  parts  of  the  spec- 
trum by  the  two  glasses,  it  is  impossible  ever  to  obtain  entire  free- 
dom from  color,  but  the  best  correction  is  obtained  when  the  eye- 
piece being  pushed  in,  the  circle  has  a  bluish  purple  exterior,  and 
when  pulled  out,  a  lemon  green  exterior.  In  the  same  way  an  un- 
der corrected  lens  should  give  inside  the  focus  a  pure  purple,  and 
outside  a  yellowish  margin ;  an  over-corrected  lens  will  give  a  blue 
or  violet  color  inside,  and  outside  an  orange  margin.  Use  the  three 
convex  lenses  in  turn,  and  note  the  colors  in  each  instance.  Many 
other  things  may  be  learnt  from  the  appearance  of  the  artificial 
star.  Thus  if  part  of  the  object-glass  is  covered,  the  circle  assumes 
the  shape  of  the  uncovered  portion.  Spherical  aberration  also 
shows  itself  by  the  formation  of  rings  in  the  image  of  unequal 
brilliancy,  and  imperfect  centering  or  obliquity  of  the  lenses,  by 
converting  the  circle  into  an  ellipse,  or  throwing  out  a  ray  of  light 
on  one  side.  This  effect  is  greatly  increased  by  using  the  mono- 
chromatic light  of  a  soda  flame. 

One  other  test  remains  to  be  applied,  that  for  imperfect  anneal- 
ing of  the  glass.  Lay  a  plate  of  glass  horizontally  in  front  of  the 
window,  so  that  the  light  reflected  from  it  shall  be  polarized.  In- 
terpose the  lens  between  it  and  the  eye,  and  examine  the  transmit- 
ted light  by  a  Nicol's  prism,  as  will  be  described  more  in  detail  in 
Experiment  88.  Any  inequality  in  density  of  the  glass  will  at 
once  show  itself  by  dark  patches,  which  change  their  position  as 
the  prism  is  turned.  Of  course  all  these  tests  must  be  re- 
garded as  secondary  to  the  real  test  of  every  large  telescope  by 
trying  it  on  various  celestial  objects  of  known  difficulty,  and  com- 
paring the  results  with  those  obtained  with  other  instruments  of 
the  same  size. 

Next  measure  the  magnifying  power  of  the  different  eye-pieces 
furnished  with  the  telescope.  The  power  of  a  small  telescope  or 


PHOTOGRAPHY.  181 

opera-glass  is  readily  measured  by  looking  simultaneously  at  the 
object  with  one  eye,  and  at  its  image  with  the  other,  and  compar- 
ing their  relative  magnitudes.  The  best  object  to  be  used  is  a 
large  scale  of  inches,  noting  how  many  divisions  as  viewed  by 
the  eye,  equal  one  or  two  as  seen  through  the  telescope.  For  high 
powers  the  best  method  is  by  the  dynameter.  Focus  the  telescope 
on  a  distant,  object,  and  turn  it  towards  the  sky,  or  other  bright 
light.  On  holding  a  sheet  of  paper  near  the  eye-piece,  a  bright 
circle  of  light  is  seen  projected  on  it,  which  is  really  an  image  of 
the  object-glass  formed  by  the  eye-piece.  The  diameter  of  the 
object-glass,  divided  by  that  of  its  image,  equals  the  magnifying 
power.  To  measure  accurately  the  diameter  of  the  small  circle,  a 
spider  line,  or  eye-piece  micrometer,  may  be  used,  or  a  small  read- 
ing microscope,  whose  objective  is  divided  in  two  parts,  which  may 
be  moved  past  each  other  a  known  amount  by  a  micrometer-screw. 
Two  images  of  the  circle  are  thus  formed,  which  may  be  rendered 
tangent  to  each  other  by  turning  the  screw.  The  parts  of  the 
objective  have  then  been  moved  a  distance  proportional  to  the 
diameter  of  the  circle,  which  is  thus  measured  with  great  preci- 
sion. In  cheap  telescopes  a  diaphragm  is  sometimes  inserted  near 
the  objective,  thus  reducing  the  available  aperture,  and  increasing 
the  sharpness  of  definition  of  a  poor  lens,  though  diminishing  the 
amount  of  light  without  apparently  reducing  its  size.  This  is 
detected  by  turning  the  eye-piece  towards  the  light,  and  seeing  if 
it  is  visible  when  looking  through  the  very  edge  of  the  objective. 
If  not,  the  diaphragm  should  be  removed  or  an  incorrect  value  of 
the  magnifying  power  will  be  obtained.  If  focussed  for  near 
objects,  the  magnifying  power  is  much  increased;  hence  for  pur- 
poses of  comparison,  objects  at  a  great  distance  should  always  be 
selected  when  making  this  measurement. 

85.     PHOTOGRAPHY.      I.  GLASS  NEGATIVES. 

Apparatus.  A  small  darkened  chamber  or  closet  is  needed  for 
this  Experiment.  In  this,  a  sink  is  placed  with  an  abundant 
supply  of  water,  and  over  it  a  shelf  for  the  bottles  containing  the 
various  re.'igents  described  below.  A  glass  or  porcelain  vessel, 
shaped  somewhat  like  a  card-case,  is  employed  to  hold  the  solution 
of  nitrate  of  silver,  and  a  flat  dish  for  the  hyposulphite  of  soda. 
A  large  number  of  plates  of  glass  are  needed  on  which  the 


182  PHOTOGRAPHY. 

photographs  are  to  be  taken,  and  racks  for  holding  them.  In 
preparing  the  solutions,  funnels,  filter-paper,  and  other  similar 
chemical  apparatus  are  also  required.  The  closet  should  be 
lighted  by  a  gas-burner  covered  with  a  yellow  shade,  to  cut  off 
the  actinic  rays.  The  camera  in  which  the  photographs  are  taken 
consists  of  a  blackened  box  with  a  convex  lens  in  front,  and  closed 
behind  by  a  plate  of  ground  glass,  or  by  the  plate-holder  which 
carries  the  prepared  plate  of  glass. 

Great  care  has  been  bestowed  on  various  forms  of  lenses  for 
cameras,  and  the  best  forms  are  somewhat  expensive.  Three 
kinds  are  commonly  employed.  First,  portrait  lenses  which  have 
a  large  aperture,  and  admitting  much  light  work  very  quickly,  but 
they  only  take  in  a  cone  with  an  angle  of  about  60°,  and  have  not 
much  depth  of  focus.  That  is,  when  focussed  for  a  given  distance, 
objects  a  little  nearer  or  a  little  farther  off  will  be  indistinct.  The 
second  kind  of  lens  is  adapted  for  views ;  a  small  concave  lens  is 
inserted  between  two  which  are  convex,  thus  giving  a  greater 
depth  of  focus,  but  not  working  so  quickly  as  the  preceding.  The 
third  class,  as  the  globe  and  the  Zentmayer  lenses,  take  in  a  large 
cone  of  light,  as  90°,  but  work  very  slowly,  requiring  one  or  two 
minutes  even  in  strong  sunlight.  They  have  a  great  depth  of 
focus,  andmay  be  placed  very  near  the  object;  but  when  this  is  a 
building,  the  perspective  will  be  bad  if  the  camera  is  brought  too 
near.  For  the  same  reason,  when  taking  a  building,  the  glass 
plate  should  always  be  vertical,  or  a  distortion  will  be  produced. 
As  no  lens  is  perfectly  achromatic,  the  focus  of  the  actinic  rays 
will  not  coincide  exactly  with  the  visual  focus,  being  less  for  an 
under,  arid  more  for  an  over-corrected  lens.  The  latter  should 
always  be  avoided,  and  it  is  best  to  get  one  in  which  the  two 
foci  coincide  as  nearly  as  possible. 

Experiment.  Almost  all  photographic  processes  depend  on 
first  taking  a  glass  negative,  that  is,  a  picture  in  which  the  bright 
portions  shall  be  transparent,  the  dark  parts,  opaque.  For  this 
purpose  the  plate  of  glass  is  first  prepared  or  rendered .  sensitive, 
then  exposed  in  a  camera  obscura  so  that  an  image  of  the  object 
shall  fall  on  it.  The  plate  is  then  developed,  by  which  the  image 
is  rendered  visible,  and  fixed  or  rendered  permanent.  These 
operations  are  easily  performed  when  all  the  apparatus  is  ready, 
and  the  baths  employed  in  good  condition.  The  real  difficulty  in 
taking  photographs,  is  in  preparing  the  various  solutions  used,  and 
in  renewing  them  as  they  deteriorate ;  accordingly  the  following 
receipts  are  given,  which  the  student  should,  if  possible,  try  for 
himself. 


PHOTOGRAPHY.  183 

The  collodion  used  for  coating  the  plates  is  made  as  follows. 
To  8  oz.  of  ether  add  96  grains  of  gun-cotton,  and  then  8  oz.  of 
alcohol.  The  strength  of  the  latter  must  be  95  per  cent.,  and  it 
must  contain  no  fusel  oil,  or  the  cotton  will  not  be  well  dissolved. 
If  made  in  the  evening  be  very  careful  about  lights,  as  the  mixture 
may  take  fire  from  a  lamp  several  feet  distant.  As  the  vapor  is 
much  heavier  than  air,  there  is  more  danger  from  lights  on  the 
floor  than  from  those  above.  Dissolve  24  grains  of  bromide  of 
potassium,  in  as  little  water  as  possible,  add  64  grains  of  iodide  of 
ammonium,  and  more  water  if  necessary,  and  pour  this  into  the 
collodion  to  iodize  it.  Pure  iodide  of  ammonium  will  do,  but  it  is 
better  to  have  the  iodine  a  little  in  excess,  in  which  case,  the  salt 
will  be  dark  colored  instead  of  white.  Too  much  iodide  and  too 
little  bromide  give  a  hard  picture.  In  about  two  days  the 
collodion  will  be  ready  for  use,  and  it  will  keep  in  good  condition 
about  two  weeks.  Iodide  of  cadmium,  which  is  commonly  used 
in  that  which  is  sold,  makes  collodion  keep  better,  but  renders  it 
less  sensitive.  If  the  weather  is  very  hot,  there  is  difficulty  from 
the  collodion  drying  too  rapidly.  In  this  case,  less  ether  and  more 
alcohol  must  be  used,  as  the  latter  is  much  less  volatile. 

The  silver-bath  by  which  the  plate  is  rendered  sensitive,  is 
made  by  adding  1  oz.  of  nitrate  of  silver  to  12  oz.  of  water,  and 
acidulating  it  with  30  or  40  drops  of  pure  nitric  acid.  If  used 
directly  it  would  dissolve  the  iodides  in  the  plate.  ^Accordingly, 
a  coated  plate  should  be  left  standing  in  it  over  night.  Filter, 
and  in  twenty-four  hours  it  can  be  used.  More  depends  on  the 
condition  of  the  silver-bath  than  on  that  of  any  other  liquid 
employed.  It  should  be  kept  nearly  neutral,  but  always  slightly 
acid.  If  alkaline  the  picture  is  fogged  or  blurred,  and  if  too  acid 
the  action  is  much  retarded.  Organic  matter  is  very  injurious, 
and  dust  should  therefore  be  carefully  excluded  by  a  cover. 
Whenever  practicable,  it  should  be  exposed  in  a  glass  bottle  to 
strong  sunlight,  which  precipitates  the  organic  matter  in  black 
flakes,  and  the  latter  removed  by  filtering.  After  using  the  bath 
for  some  time,  it  becomes  covered  with  a  scum,  due  to  the  alcohol 
and  ether  from  the  collodion  plate,  which  is  the  first  indication 
that  the  solution  is  becoming  too  weak.  The  iodide  of  silver  is 
then  liable  to  be  precipitated  on  the  plate,  forming  little  spots  in 


184  PHOTOGRAPHY. 

it  like  pinholes.  One  half  its  bulk  of  water  should,  in  this  case, 
be  added,  to  precipitate  the  iodide,  the  solution  filtered  and  boiled 
down  to  its  original  strength. 

To  develope  the  picture  when  taken  from  the  camera,  a  solution 
of  1  oz.  of  proto-sulphate  of  iron,  in  about  20  oz.  of  water  is  used, 
and  containing  1  or  2  oz.  of  pure  acetic  acid,  No.  8.  This  liquid 
will  keep  only  about  three  days.  The  object  of  the  acetic  acid  is 
to  retard  the  'process,  as  otherwise  the  silver  would  blacken 
instantly. 

The  liquid  used  to  fix  the  picture,  is  formed  by  dissolving  1  oz. 
of  hyposulphite  of  soda  in  5  oz.  of  water. 

Be  careful  that  the  glass  is  not  rusty  or  iridescent,  as  in  that  case 
the  collodion  is  liable  to  cleave  off.  Double  thick  glass  is  prefer- 
able on  account  of  its  greater  strength,  unless  the  plates  are 
small.  There  must  be  no  "  knobs  "  or  glass  dust  on  its  surface,  nor 
deep  scratches,  because  these  will  appear  in  the  positive  if  the 
printing  is  done  by  sunlight.  Almost  all  plates  are  slightly 
curved,  and  it  is  essential  that  the  hollow  or  concave  side  should  be 
coated.  The  plate  will  then  conform  more  nearly  to  the  image  in 
the  camera,  is  easier  to  coat,  is  less  liable  to  be  scratched  if  laid 
down  on  its  face,  and  is  not  so  likely  to  be  broken  in  printing. 

To  clean  the  glass,  mix  equal  parts  of  alcohol  and  ammonia,  and 
add  enough  rotton-stone  to  render  it  viscid.  Pour  a  few  drops  on 
the  glass  and  scour  with  wash  leather,  letting  the  plate  rest  on  one 
corner.  Clean  also  the  edges  carefully.  Let  it  dry,  and  then  rub 
off  all  the  rotton-stone  with  clean  flannel.  Breathe  on  the  plate, 
and  if  clean,  the  moisture  will  pass  off  rapidly  and  evenly  from 
the  surface.  Thus  cleaned,  they  will  keep  for  two  days.  A 
better  method  is  the  following.  Soak  the  plates  over  night  in 
strong  nitric  acid,  and  wash  thoroughly  under  a  faucet.  Mix  up 
the  white  of  an  egg  with  an  egg-beater,  taking  care  that  none  of 
the  yolk  gets  in,  add  12  ounces  of  water  and  3  or  4  drops  of 
concentrated  ammonia,  which  keeps  the  egg  from  souring,  and 
neutralizes  any  acid  that  may  remain  on  the  plate.  Filter  through 
paper  or  cloth  and  keep  the  filtrate  in  a  bottle.  Pour  it  over  the 
plate,  and  a  uniform  film  is  produced,  which  will  last  for  six 
months  if  kept  dry  and  free  from  dust.  Another  method  is  to 
soak  the  glass  over  night  in  a  strong  solution  of  caustic  potash 


PHOTOGRAPHY.  185 

and  then  wash  it,  but  this  is  liable  to  injure  the  silver  bath.  If 
any  nitric  acid  gets  in,  it  merely  retards  the  action,  but  alkali  will 
cause  fogging,  or  the  image  will  look  smoky,  as  if  under  a  veil. 

To  coat  a  plate  properly  with  collodion  requires  considerable 
skill  and  practice,  as  it  is  very  essential  that  the  coating  should  be 
perfectly  uniform.  First,  remove  any  films  of  collodion  that  may 
have  dried  on  the  mouth  of  the  bottle,  and  take  care  never  to 
disturb  the  sediment  at  the  bottom.  Regarding  the  plate  as  a 
map,  it  is  held  by  the  &.  W.  corner  with  the  thumb  and  forefinger 
of  the  left  hand,  and  the  collodion  poured  on  just  N".  E.  of  the 
centre.  Enough  is  added  to  cover  about  half  the  plate  which  is 
then  inclined,  so  that  the  liquid  shall  flow  successively  to  the 
N.  E.,  the  N~.  TK,  the  &.  Wl  and  the  /&  E.  corners,  and  then  tipped 
so  that  it  will  run  off  into  the  bottle.  Allow  it  to  drain  for  a  few 
seconds  and  incline  it  gently  from  side  to  side,  to  prevent  its 
forming  streaks.  The  ether  soon  evaporates,  and  as  soon  as  the 
film  becomes  sticky  and  consistent,  the  plate  is  immersed  in  the 
silver  bath,  by  laying  it  on  its  holder,  and  lowering  it  into  the 
liquid.  This  must  be  done  slowly  and  steadily,  or  streaks  will 
appear  across  the  plate.  If  the  plate  is  too  large  to  be  held  by 
one  corner  while  coating  it,  lay  it  on  the  palm  of  the  hand,  and 
interpose  a  sheet  of  card-board  to  prevent  the  warmth  from  drying 
the  collodion  too  rapidly.  Still  larger  plates  must  be  placed  on  a 
board  and  rested  on  a  point  attached  to  the  top  of  a  tripod. 

After  remaining  a  few  seconds  in  the  bath,  the  plate  should  be 
raised  gently  out  of  it,  when  its  surface  will  present  a  greasy 
appearance,  due  to  the  ether  still  remaining  in  the  film ;  soon 
however,  it  will  appear  to  be  perfectly  wet  by  the  solution,  and  is 
then  ready  to  be  transferred  to  the  camera.  The  plate-holder 
consists  of  a  frame  closed  in  front  by  a  slide,  and  with  a  hinged 
back.  The  latter  is  opened,  the  plate  put  in  with  the  coated  side 
next  the  slide,  and  the  latter  then  closed.  A  black  cloth  is  thrown 
over  it  to  cut  off  any  stray  light  that  may  leak  in  through  cracks 
in  the  holder,  and  it  is  then  carried  to  the  camera.  The  object  to 
be  taken  having  been  placed  in  the  proper  position,  and  in  a  good 
light,  the  camera  is  turned  towards  it,  when  an  image  will  be 
formed  on  the  ground  glass.  This  is  best  seen  by  standing  behind 
the  instrument,  and  cutting  off  the  light  by  a  black  cloth  thrown 


186  PHOTOGRAPHY. 

over  the  head.  The  size  of  the  image  may  be  varied  by  altering 
the  distance  of  the  camera,  and  the  latter  focussed  by  changing 
the  distance  between  the  ground  glass  and  lens.  For  near  objects 
the  focus  is  greater  and  the  size  of  the  image  larger  than  for 
those  more  distant.  The  image  may  be  brought  to  the  centre 
of  the  glass,  by  turning  the  camera  or  inclining  it.  When  the 
image  is  satisfactory  and  carefully  focussed,  remove  the  ground 
glass,  and  replace  it  by  the  holder.  Cover  the  lens  with  a  cap  or 
black  cloth.  Then  draw  the  slide  of  the  holder,  and  when  all  is 
ready,  expose  the  plate  to  the  light  by  removing  the  cap.  The 
time  of  exposure  varies  with  the  light,  the  object,  arid  the  kind 
of  lens,  and  must  be  learned  by  experience.  When  the  time  has 
expired,  replace  first  the  cap,  and  then  the  slide,  and  taking  out 
the  holder  carry  it  into  the  dark  room.  Take  care  and  never  turn 
the  plate-holder  over  after  the  plate  is  in  it,  as  the  silver  collects 
along  the  low.er  edge  and  would,  if  inverted,  flow  over  the  glass, 
forming  streaks. 

On  taking  the  plate  out  of  the  holder  no  trace  of  the  picture  is 
visible,  but  the  film  merely  appears  of  a  creamy  white  throughout. 
Pour  on  some  of  the  developer  so  as  to  cover  the  plate  at  one 
flow  (or  streaks  will  be  formed),  holding  the  glass  horizontal  with 
the  prepared  side  up,  and  in  a  few  seconds  the  picture  will  appear, 
the  portions  acted  on  by  light  turning  dark,  the  others  remaining 
unchanged.  The  plate  is  then  washed  under  the  faucet  to  remove 
the  developer.  If  the  exposure  has  been  too  long,  the  picture  will 
develop  instantly,  giving  a  dense  blurred  negative.  With  too 
short  an  exposure  or  too  feeble  a  light,  a  faint  transparent  picture 
is  obtained,  which  developes  only  very  slowly.  In  this  case,  it 
may  be  improved  by  re-developing,  or  pouring  on  a  mixture  of 
the  silver  bath  and  developer  as  when  developing.  Pryrogallic 
acid  is  sometimes  used  instead  of  sulphate  of  iron. 

The  picture  is  then  rendered  permanent  by  immersing  it  in  the 
solution  of  hyposulphite  of  soda,  which  dissolves  the  iodide  of 
of  silver,  where  unacted  on  by  light,  rendering  these  parts  of  the 
plate  transparent. 

It  is  still  easily  scratched,  and  should  be  varnished  if  it  is  to  be 
handled  much.  For  this  purpose,  it  is  warmed  over  a  lamp  until 
hot  to  the  touch,  and  amber  varnish  poured  on  precisely  as  when 


PHOTOGRAPHY.  187 

coating  a  plate.  If  not  warmed,  the  varnish  gives  a  precipitate. 
Be  careful  that  it  does  not  catch  fire,  as  it  then  dries  in  ridges. 
Warm  again  gently,  to  harden  the  varnish.  The  completed 
negatives  are  best  kept  side  by  side  and  vertical,  in  racks. 

The  most  common  difficulty  in  taking  photographs  is  fogging, 
or  the  picture  appearing  misty  or  indistinct.  In  this  case,  test  the 
silver  bath,  and  if  it  is  alkaline  add  a  little  nitric  acid.  If  too 
much  is  added  the  process  is  retarded,  and  it  must  be  nearly 
neutralized  with  ammonia.  Great  care  must  be  taken  that  the 
plate  is  not  exposed  to  stray  light  before  or  after  placing  it  in  the 
camera,  and  that  the  latter  and  the  plate-holder  have  no  cracks  in 
them,  through  which  the  light  may  enter. 

When  taking  views,  the  camera  should  never  face  the  sun  if  it 
can  possibly  be  avoided.  If  this  is  unavoidable,  hold  a  hat  or 
board  so  that  its  shadow  shall  fall  on  the  camera,  thus  cutting  off 
the  direct  rays  of  the  sun  from  the  lens.  When  light  and  dark 
objects,  as  the  sky  and  deep  shadows,  are  to  be  taken  at  the 
same  time,  hold  a  screen  in  front  of  the  lens,  to  cut  off  the  brighter 
objects  until  near  the  end  of  the  exposure,  and  then  remove  it. 

As  objects  for  the  stereoscope  or  lantern,  glass  positives  are 
needed,  that  is,  photographs  on  glass,  in  which  the  light  portions 
shall  be  transparent,  and  the  dark  parts  opaque.  A  negative  is 
first  taken  and  then  photographed  like  any  other  object,  reflecting 
the  light  of  the  sky  through  it  by  a  sheet  of  paper,  and  cutting  off 
all  stray  light  in  front.  Porcelain  photographs  were  formerly 
taken  in  this  way,  but  both  are  now  often  printed  like  paper  posi- 
tives. Ambrotypes  are  negatives  exposed  for  a  short  time,  so  that 
the  dark  portions  are  very  transparent,  and  are  rendered  black  by 
placing  a  sheet  of  tin  covered  with  black  varnish  behind  them. 

86.     PHOTOGRAPHY.      II.  PAPEB  POSITIVES. 

Apparatus.  Some  albumenized  paper,  and  the  reagents  needed 
to  render  it  sensitive  and  to  make  the  picture  permanent.  Also 
some  negatives  to  be  copied,  and  several  printing-frames  in  which 
to  expose  the  paper  to  the  light.  Strong  sunlight  is  requisite  for 
this  Experiment,  and  it  is  much  better  to  place  the  printing-frames 
outside  the  window,  rather  than  let  the  sunlight  first  pass  through 
the  glass.  Three  flat  porcelain  dishes  are  needed  to  hold  the 


188  PHOTOGRAPHY. 

liquids  in  which  the  paper  is  immersed,  spring  clothes-pins  by 
which  to  hold  it  when  drying,  and  an  abundant  supply  of  water 
in  which  to  wash  it. 


Experiment.  It  is  difficult  to  judge  of  the  true  appearance  of 
an  object  from  the  glass  negative,  as  the  lights  and  shades  are 
inverted,  and,  moreover,  several  copies  of  a  photograph  are  often 
wanted  for  distribution.  It  is  therefore  customary  to  print 
positives  on  paper,  by  the  process  described  below.  A  fine  grained 
paper  is  employed,  covered  with  a  thin  layer  of  albumen,  which 
fills  up  the  pores  and  gives  a  good  surface.  Excellent  albumen- 
ized  paper  may  now  be  procured  ready-made,  as  it  is  manufac- 
tured on  a  large  scale,  for  photographic  purposes.  When  the 
picture  is  to  be  touched  up  with  India  ink,  or  colored,  common 
paper  is  often  preferred,  as  the  paint  will  not  adhere  to  the 
albumen.  To  render  it  sensitive,  a  solution  of  nitrate  of  silver  is 
prepared,  of  a  strength  of  about  40  grains  to  the  ounce,  but 
depending  on  the  amount  of  chloride  of  sodium  in  the  paper.  It 
should  also  be  somewhat  stronger  in  summer  than  in  winter. 
Render  it  alkaline  by  a  few  drops  of  ammonia,  and  pour  it  into  a 
flat  porcelain  dish.  If  any  scum  appears  on  the  surface,  remove  it 
by  a  piece  of  tissue  paper. 

Take  a  piece  of  the  paper,  somewhat  smaller  than  the  dish,  by 
the  two  opposite  edges,  with  the  albumenized  side  down,  and  dip 
it  into  the  dish  so  that  the  centre  line  shall  touch  the  liquid. 
Lower  the  edges  so  that  the  paper  shall  float  on  the  liquid,  taking 
care  that  no  air-bubbles  are  imprisoned  under  it,  or  they  will  form 
white  circles  on  the  picture.  Breathe  on  the  edges  of  the  paper 
to  prevent  their  curling  up,  by  the  warping  due  to  the  expansion 
of  the  lower  side,  and  take  great  care  that  no  silver  gets  on  the 
upper  side  of  the  sheet.  The  time  of  immersion  must  be  found 
by  trial ;  it  is  generally  one  or  two  minutes.  If  too  long,  the  sil- 
ver works  through  the  albumen  into  the  paper,  turning  it  yellow 
on  exposure,  while  if  too  short,  the  salt  does  not  decompose  and 
fill  up  the  surface  with  the  silver,  so  that  a  mottled  appearance  is 
produced.  To  obtain  the  best  results,  various  devices  are  em- 
ployed, such  as  fuming  the  paper  with  ammonia,  which  gives  a 
clearer  picture.  After  taking  the  paper  out  of  the  bath  it  should 


PHOTOGRAPHY.  189 

be  hung  up  by  the  corners  by  spring  clothes-pins,  in  the  dark.  It 
will  only  keep  a  short  time,  in  winter  for  a  day  or  two,  in  summer 
scarcely  over  night.  The  prepared  surface  should  never  be 
touched  by  the  fingers,  or  their  mark  will  appear  in  the  finished 
picture. 

The  paper  must  now  be  exposed  to  sunlight  under  the  negative. 
The  pressure  frame  in  which  this  is  effected  is  made  somewhat  like 
the  frame  of  a  picture,  only  it  is  much  more  substantial,  and  so 
arranged  that  the  back  can  be  removed;  the  latter,  when  in 
place,  is  pressed  strongly  against  the  glass  in  front,  by  springs. 
The  back  is  lined  with  felt,  and  hinged,  so  that  one  half  can  be 
turned  back,  and  part  of  the  picture  examined  without  disturbing 
the  remainder.  Having  cut  the  paper  of  the  right  size,  the  pres- 
sure-frame is  placed  face  downwards,  and  the  back  removed.  A 
plate  of  glass  is  first  inserted,  like  the  glass  of  a  picture,  then  the 
negative  with  the  prepared  side  up,  next  the  paper  with  the  al- 
bumenized  side  down,  and  finally  the  back  of  the  frame  is  restored 
to  its  place.  Great  care  must  be  taken  always  to  place  the  pre- 
pared sides  of  the  glass  and  paper  in  contact,  as  otherwise  only  a 
blurred  picture  will  be  obtained. 

Now  turn  the  frame  over,  and  place  it  in  strong  sunlight,  and 
inclined  at  such  an  angle  that  the  light  shall  fall  on  it  nearly  nor- 
mally. It  will  soon  be  seen  that  the  unprotected  portions  of  the 
paper,  that  is,  those  under  the  transparent  parts  oi  the  negative, 
are  turning  under  the  influence  of  sunlight  from  white  to  black. 
After  a  few  minutes,  take  the  frame  out  of  the  light,  and  opening 
half  the  back,  bend  the  paper  so  as  to  examine  it,  when  the  pic- 
ture will  appear  in  its  true  aspect  with  shades  dark,  and  the  bright 
portions,  light.  If  the  picture  is  not  dark  enough,  replace  the  back, 
and  thus  proceed,  until  the  paper  is  somewhat  darker  than  is 
desired  for  the  finished  photograph.  The  picture  thus  obtained  is 
of  a  purplish  color,  and  if  exposed  to  light  would  turn  completely 
dark.  It  must  therefore  first  be  toned,  or  its  color  altered,  and 
then  fixed,  or  rendered  permanent. 

The  bath  for  toning  is  made  of  a  solution  of  chloride  of  gold. 
Procure  some  pure  gold  from  a  gold  beater,  dissolve  it  in  aqua 
regia  and  evaporate  to  dryness.  Dissolve  5  grains  of  the  yellow 
chloride  of  gold  thus  obtained,  in  1  oz.  of  water,  and  neutralize  it 


190  PHOTOGRAPHY. 

with  carbonate  of  soda.  This  must  then  be  mixed  with  about  ten 
times  its  bulk  of  water,  using  a  stronger  solution  in  cold  than  in 
warm  weather.  Wash  the  picture  thoroughly  in  running  water 
for  five  or  ten  minutes,  to  remove  all  the  free  silver  which  would 
otherwise  precipitate  the  gold  and  give  a  foggy  picture,  and  then 
dip  it  face  downwards  into  the  toning  solution.  The  effect  of  this 
will  be  to  alter  its  color  until  a  certain  reddish  blue  tint  is  attained. 
The  best  effects  are  obtained  with  a  lukewarm  solution,  of  such  a 
strength  as  to  tone  in  about  ten  minutes.  If  not  toned  sufficiently 
the  color  is  reddish,  and  if  this  process  is  carried  too  far  it  takes 
the  life  out  of  a  picture,  and  gives  it  a  chalky  appearance.  In  the 
same  way,  too  rapid  a  toning  gives  a  mealy  look  to  the  photo- 
graph. As  the  paper  withdraws  the  gold  gradually  from  the  solu- 
tion, rendering  it  weaker,  more  should  be  added,  first  neutralizing 
it  with  soda.  It  is  well  to  move  the  bath  gently,  swaying  it  from 
side  to  side.  Be  very  careful  that  no  hyposulphite  of  soda  gets 
into  the  gold  solution,  as  it  would  spoil  it,  forming  a  precipitate. 
As  the  paper  is  still  sensitive  to  light,  the  toning  should  be  carried 
on  in  a  dark  room,  just  light  enough  to  distinguish  clearly  the 
color  attained.  To  stop  the  toning,  the  photograph  is  next  washed 
in  cold  water  to  remove  the  gold ;  it  is  then  rendered  permanent 
by  immersion  in  a  solution  of  hyposulphite  of  soda,  1  oz.  of  the 
salt  to  8  oz.  of  water.  As  the  picture  is  rendered  lighter  and  red- 
der by  this  process,  allowance  must  be  made,  in  the  printing  and 
toning.  The  photograph  must  now  be  washed  very  thoroughly  in 
running  water  for  an  hour  or  two,  or  better,  over  night,  to  remove 
every  trace  of  hyposulphite.  Otherwise  a  black  sulphide  of  silver 
is  formed,  which  turns  in  time  to  red,  and  spoils  the  picture.  To 
see  if  all  the  hyposulphite  is  removed,  hold  the  paper  up  to  the 
light,  when  it  should  appear  clear ;  if  mottled,  the  washing  must 
be  continued. 

It  now  only  remains  to  mount  the  photograph,  that  is,  to  paste  it 
on  to  cardboard,  to  make  it  flat  and  stiff.  Lay  it  face  upwards  on 
a  plate  of  glass,  and  lay  on  it  a  second  plate,  of  the  size  to  which 
it  is  to  be  cut.  It  is  thus  easily  centered,  and  cut  to  the  right  size 
by  drawing  a  sharp  knife  along  the  edges  of  the  glass,  holding  the 
blade  always  at  the  same  angle  of  inclination.  Attach  it  to  the 
cardboard  by  paste  made  of  the  best  wheat  starch.  The  latter  is 


TESTING   THE    EYE.  191 

first  moistened  with  cold  water,  and  boiling  water  then  poured 
upon  it ;  if  it  does  not  come  thick  enough  it  may  be  heated  for  a 
moment  to  the  boiling  point,  but  should  not  be  boiled,  otherwise 
it  becomes  watery  and  loses  its  adhesiveness.  To  prevent  the 
cardboard  from  curling  up,  it  is  well  to  moisten  it  before  attaching 
the  photograph.  The  whole  should  then  be  passed  through  heavy 
rollers  to  give  a  good  finish  to  the  picture. 

87.    TESTING    THE  EYE. 

Apparatus.  A  set  of  test-types,  or  letters  of  various  sizes,  should 
be  placed  at  distances  from  the  student  proportional  to  their  size. 
On  the  table  are  placed  the  optometers  described  below,  a  reading 
microscope  on  a  stand,  for  the  experiment  of  Cramer  and  Helm- 
holtz,  a  gas  flame,  tests  for  astigmatism,  and  a  set  of  concave, 
convex  and  cylindrical  lenses  of  various  curvatures,  and  prisms  of 
various  angles. 

Experiment.  The  eye  is  formed  like  a  camera  obscura,  in 
which  the  retina  takes  the  place  of  the  screen  on  which  the  image 
is  received.  In  front  of  the  lens  is  a  delicate  curtain,  called  the 
iris,  which  gives  to  the  eye  its  color,  and  in  this  is  a  circular  hole, 
the  pupil.  The  iris  is  formed  of  fibres,  some  circular,  others 
radial,  the  contraction  of  the  first  diminishing,  of  the  second 
increasing  the  size  of  the  pupil,  and  hence  the  amount  of  light 
admitted  into  the  eye.  These  changes  are  readily  seen  by  cover- 
ing the  eye  with  the  hand,  removing  the  latter,  and  looking  in  a 
mirror;  the  pupil  will  then  be  seen  to  contract  slowly,  having 
dilated  in  the  dark.  The  pupil  of  the  other  eye  will  also  contract 
a  little,  as  they  both  commonly  act  together. 

The  image  of  objects  at  various  distances,  may  be  brought  to  a 
focus  on  the  retina  by  varying  the  form  of  the  lens,  while  in  the 
camera  it  is  effected  by  varying  its  position.  By  this  change 
which  is  called  accommodation,  objects  may  be  seen  with  perfect 
distinctness,  with  a  normal  or  perfect  eye,  at  any  distance,  from 
about  4  inches  to  infinity.  Call  P  the  nearer  distance,  and  R  the 

farther,  for  any  eye,  then  ~7  —  "p  —  ~R   '1S  ca^e<^  tne  range   of 

accommodation,  and  is  much  employed  in  studying  defects  of  the 
eye.  In  the  case  of  the  normal  eye,  the  range  of  accommodation 


192  TESTING    THE    EYE. 

evidently  equals  £.  The  most  common  defect  to  which  the  eye  is 
subject,  is  that  the  ball  is  not  spherical.  If  it  is  elongated,  the 
retina  is  carried  too  far  off,  and  objects  must  be  brought  nearer 
the  eye,  to  render  them  distinctly  visible.  Such  an  eye  is  called 
myopic,  or  near-sighted.  If  the  ball  is  flattened,  near  objects 
cannot  be  easily  seen,  and  the  eye  is  then  hypermetropic,  or  far- 
sighted.  This  must  not  be  confounded  with  the  effect  of  age, 
which  renders  the  lens  harder  and  thus  diminishes  the  range  of 
accommodation,  so  that  distant  objects  alone  can  be  seen.  The 
eye  is  then  said  to  be  presbyopic.  The  normal  eye  is  called 
emmetropic. 

To  measure  the  far  and  near  points  optometers  are  used.  One 
of  the  simplest  of  these  consists  of  a  board  on  which  a  straight 
line  is  ruled.  At  one  end  is  a  sheet  of  metal,  with  two  fine  slits 
very  near  together.  The  eye  is  placed  close  to  the  slits,  so  as  to 
look  through  both,  when  it  will  be  noticed,  that  the  nearer  end  of 
the  line  appears  double,  since  the  images  formed  by  the  two  slits 
cannot  be  brought  together  by  the  eye,  on  account  of  the  short 
distance.  Sometimes,  also,  the  farther  end  will  appear  double,  if 
the  eye  is  myopic.  The  points,  where  the  line  divides,  give  the 
far  and  near  limits  of  distinct  vision.  A  better  form  of  opto- 
meter,  resembles  the  apparatus  represented  in  Fig.  60,  only  it  is 
much  smaller.  The  lens,  which  should  have  a  focus  of  6  inches,  is 
fixed  at  the  end  of  the  rod,  in  the  place  of  the  gas-burner,  A,  and 
some  very  fine  print,  or  other  minute  object,  is  attached  to  the 
screen,  C.  Now  measure  the  greatest  and  least  distance  at  which 
the  print  can  be  read,  when  the  eye  is  placed  near  the  lens.  Call 

these  distances  Pf  and  JRf.     Then  -p  =  -p-  —  -g-,  and    j?  =  ^ 

—  -g-,  which  gives  P  and  _K,  the  far  and  near  distances  of  accom- 
modation. For  the  normal  eye,  as  P  =  10,  R  —  oo ,  P'  should 
equal  2.67,  and  E  =  6. 

Another  excellent  form  of  optometer  is  very  simply  made  of  a 
sheet  of  cardboard  or  brass.  This  is  pierced  with  three  sets  of 
holes,  the  first  a  single  hole  1  mm.  in  diameter,  the  second  several 
smaller  holes  near  together,  for  instance  three  rows  of  three  each 
at  intervals  of  a  millimetre,  and  thirdly  two  holes  3  mm.  apart, 


TESTING    THE    EYE.  193 

one  of  which  may  be  covered  with  a  plate  of  red  glass.  View  a 
small  distant  point  of  light,  as  a  candle  or  gas  flame,  through 
them,  and  the  appearance  will  vary  according  as  the  eye  is  normal, 
far  or  near  sighted.  Looking  through  the  single  hole  and  moving 
the  card  rapidly  from  side  to  side,  the  light  will  appear  to  remain 
stationary,  if  the  eye  is  normal,  otherwise  it  will  appear  to  move 
as  the  rays  pass  through  different  portions  of  the  pupil.  In  the 
same  way  the  nine  holes  will  give  nine  images,  if  the  eye  is  not 
normal.  If  the  two  holes  are  used,  two  circles  are  seen  which 
overlap  as  the  card  is  brought  near  the  eye.  If  the  eye  is  not 
normal  two  images  will  be  formed  in  the  overlapping  part,  since 
the  rays  falling  on  different  parts  of  the  lens  are  not  brought 
together  to  the  same  point  on  the  retina.  Now  cover  the  right 
hand  hole  with  the  red  glass,  and  if  the  eye  is  far-sighted  the  left 
hand  one  will  be  colored,  if  near-sighted  the  right  one,  since  in 
the  latter  case  the  rays  cross,  coming  to  a  focus  before  reaching 
the  retina.  From,  the  distance  between  the  images,  the  amount 
of  the  defect  may  be  measured.  Thus  bring  a  second  candle 
flame  near  the  first,  until  two  of  the  images  overlap,  forming  three 
instead  of  four ;  the  distance  between  the  candles  then  equals  the 
interval  between  the  images,  and  from  it  the  lens  required  to 
render  the  eye  normal,  may  be  determined.  Let  D  be  the  interval 
between  the  images,  d  that  between  the  two  holes,  and  B  the 
distance  from  the  light  to  the  eye.  Then,  D  :  d  =-•  JB  : '  f  in  which 
f  is  the  focal  length  of  the  lens  required  to  produce  distinct 
vision.  By  turning  the  card  the  twro  images  will  appear  to  re- 
volve around  each  other,  and  if  the  eye  is  astigmatic  their  distance 
apart  will  vary.  If  the  eye  is  normal  all-  these  effects  may  still 
be  observed  by  putting  on  convex,  concave,  or  cylindrical  glasses. 
When  observing  one's  own  eye  it  is  often  more  convenient  to 
view  the  reflection  of  two  lights  near  at  hand  in  a  distant  mirror, 
so  that  their  distance  apart  may  be  more  easily  varied. 

The  best  test,  however,  for  the  eye,  is  to  see  if  all  the  test-letters 
can  be  read  easily.  To  understand  how  objects  look  to  a  near- 
sighted person,  put  on  a  pair  of  convex  glasses,  and  repeat  these 
observations  with  them.  Do  the  same  with  concave  glasses,  which 
give  the  effect  of  hypermetropia.  See  also  if  the  foci  of  the  glasses 
can  be  determined  correctly  from  these  observations. 

13 


194  TESTING    THE    EYE. 

Another  defect  present  in  some  eyes  is  astigmatism,  or  unequal 
focus  for  horizontal  and  vertical  lines.  For  example,  the  eye  may 
be  normal  for  vertical,  and  near-sighted  for  horizontal  lines.  It 
is  detected  by  looking  at  a  test  made  of  several  series  of  strongly 
marked  equidistant  lines,  running  in  various  directions.  This 
defect  is  corrected  by  using  cylindrical  lenses,  or  if,  as  often  hap- 
pens, the  eye  is  myopic  or  hypermetropic  at  the  same  time,  by 
means  of  lenses  cylindrical  on  one  side,  and  convex  or  concave  on 
the  other.  Many  persons  who  could  never  see  well  with  common 
glasses,  experience  wonderful  relief  from  such  lenses.  Sometimes 
the  axes  of  the  eyes  are  not  quite  parallel,  a  defect  remedied  by 
the  aid  of  prisms,  with  very  acute  angles. 

Many  theories  have  been  advanced  to  explain  accommodation, 
some  supposing  that  the  retina  was  drawn  back,  others,  that  the 
lens  moved,  and  others,  that  the  ball  of  the. 
eye  changed  its  shape.     The  true  explana- 
tion is  deduced  from  the  following  experi- 
ment, devised  and  worked  out  quantitively, 
by  Kramer  and  Helmholtz.      Two  persons 
m    62  are  required;  one,  whose  eye  is  to  be  exam- 

ined,  sits   facing   a   candle,   or    gas-burner, 

while  the  other  examines  with  the  reading  microscope  the  reflec- 
tion of  the  light  in  his  eye.  Three  images  will  be  seen,  as  shown 
in  Fig.  62,  in  which  V  ig  intended  to  represent  the  reflection  of 
the  candle  flame.  The  eye  being  directed  towards  a  distant  ob- 
ject, the  first  image  to  the  right  is  formed 
by  reflection  in  the  cornea,  or  front  surface 
of  the  eye.  It  is  bright  and  upright,  as  the 


V 


v 


V 


Fig.  63. 


\  /  surface  is  convex.  The  second  is  formed 
by  the  front  surface  of  the  lens.  It  is  much 
fainter  and  larger,  but  also  upright.  The 
third  being  formed  in  the  posterior  and 
concave  surface  of  the  lens,  is  minute  and  inverted.  Now  let 
the  eye  be  directed  towards  a  near  object.  The  first  and  third 
images  will  remain  unchanged  both  in  size  and  position,  showing 
that  the  cornea  and  rear  surface  of  the  lens  are  not  altered,  either 
in  position  or  curvature.  But  the  second  image,  as  shown  in  Fig. 


TESTING    THE    EYE.  195 

63,  approaches  the  first,  and  diminishes  in  size,  showing  that  the 
front  surface  of  the  lens  is  pushed  forward,  and  becomes  more 
curved.  Measurements  also  show,  that  the  amount  of  the  change 
is  just  sufficient  to  account  for  the  required  difference  in  focus. 
This  experiment  is  very  conclusive,  as  each  of  the  other  hypoth- 
eses is  disproved  by  it.  If  the  cornea  altered,  the  first  image 
only  should  move.  If  the  lens  moved,  the  second  and  third 
images  should  approach  the  first  without  altering  their  size,  and 
if  the  form  of  the  ball  altered,  the  relative  position  of  all  three 
should  remain  unchanged. 

All  parts  of  the  retina  are  not  equally  sensitive ;  although  the  eye 
can  perceive  objects  through  an  angle  of  about  150°  horizontally, 
and  120°  vertically,  yet  the  portion  where  vision  is  most  distinct 
is  quite  small,  not  more  than  3°  or  4°  in  diameter.  This 
portion  of  the  retina,  which  is  called  the  macula  lutea,  is  used 
almost  exclusively  whenever  objects  are  carefully  examined,  and 
probably  on  this  account,  is  not  quite  as  sensitive  to  very  faint 
objects  as  the  adjacent  parts.  At  any  rate,  it  is  very  customary 
with  astronomers,  when  trying  to  see  very  faint  objects,  to  direct 
the  eye  a  short  distance  from  their  supposed  place,  and  try  to 
catch  sight  of  them,  when  not  in  the  centre  of  the  field  of  view. 
A  short  distance  from  the  macula  lutea,  on  the  side  towards  the 
nose,  is  a  small  circle  where  the  optic  nerve  enters.  This  space, 
although  so  near  the  most  sensitive  portion  of  the  retina,  is 
totally  insensible  to  light.  It  is  called  the  papilla,  or  some- 
times the  punctum  ccecum,  or  blind  spot.  To  observe  it,  mark 
two  points  on  a  sheet  of  paper,  about  4  inches  apart,  and  closing 
the  left  eye,  direct  the  other  to  the  left  hand  point,  and  then 
moving  the  paper  to  and  fro,  a  certain  distance  will  be  found,  at 
which  the  other  point  will  completely  disappear.  By  using  two 
lights,  this  experiment  maybe  rendered  still  more  striking,  as  even 
a  bright  light  may  be  made  to  completely  disappear,  although  ob- 
jects all  around  it  are  visible. 

A  great  variety  of  experiments  may  be  made,  depending  on  the 
stereoscopic  effects  obtained  with  two  eyes,  or  on  the  persistence 
of  vision,  using  such  instruments  as  the  thaumatrope,  chromatrope, 
and  phenakistascope. 


196  OPTHALMOSCOPE. 

88.     OPTHALMOSCOPE. 

Apparatus.  The  instrument  known  as  the  "  Opthalmoscopic  Eye 
of  Dr.  Perrin,"  is  admirably  adapted  as  a  substitute  for  a  human 
eye,  on  which  to  use  the  opthalmoscope.  It  consists  of  a  brass 
ball,  on  a  stand,  representing  the  globe  of  the  eye,  a  series  of  cups, 
painted  to  represent  different  diseases  of  the  retina,  which  may  be 
inserted  in  its  rear  portion,  and  lenses,  representing  the  cornea  and 
lens,  which  may  be  screwed  on  in  front.  To  these,  diaphragms, 
representing  the  change  in  diameter  of  the  pupil,  or  aperture  in 
the  iris,  may  be  attached.  An  Argand  burner  is  needed,  and  a 
Grafe's  opthalmoscope,  also  some  plates  of  glass,  and  a  small  mir- 
ror, with  the  silvering  removed  from  a  circle  a  quarter  of  an  inch 
in  diameter.  This  Experiment  should  be  performed  in  a  darkened 
room,  but  instead,  the  light  may  be  cut  off  by  a  large  screen 
of  black  cloth. 

Experiment.  The  opthalmoscope  which  is  used  in  studying  the 
interior  of  the  eye,  has  caused  a  complete  revolution  in  this  branch 
of  medical  science.  Its  inventor,  Helmholtz,  reflected  light  into 
the  eye,  by  a  piece  of  plate  glass,  and  then  looking  through  it, 
found  the  interior  sufficiently  illuminated  to  be  visible. 

Take  the  model  of  the  eye  from  its  box  and  place  it  on  its 
stand.  Insert  the  retina  marked  1,  which  represents  the  normal 
or  healthy  retina.  Screw  on  in  front  the  lens  marked  E.  J/.,  and 
light  the  burner,  placing  it  by  the  side  of  the  model  eye,  and 
about  a  foot  distant.  By  reflecting  the  light  into  the  eye  by 
a  plate  of  glass,  and  looking  through  the  latter,  Helmholtz' 
experiment  may  be  repeated,  and  a  view  of  the  interior  obtained. 
This  is  more  easily  accomplished  by  using  several  plates,  or  better 
still,  with  the  mirror  from  which  the  silvering  has  been  partially 
removed. 

The  opthalmoscope  consists  of  a  circular,  concave  mirror,  with 
the  silvering  removed  from  the  centre.  Just  behind  it  is  placed 
a  fork,  in  which  either  of  the  five  small  lenses  may  be  placed. 
They  should  be  numbered  on  their  edges  from  1  to  5,  the 
former  being  the  most  concave,  the  latter  the  most  convex.  The 
retina  of  the  normal  eye  is  placed  at  a  distance  from  the  lens, 
equal  to  its  principal  focus,  hence  its  image  is  formed  at  an 
infinite  distance,  or  the  rays  emerge  parallel.  It  can  therefore  be 
viewed  without  any  lens,  using  the  mirror  precisely  as  in  the 


OPTHALMOSCOPE.  197 

previous  experiment.  The  image  is  better  seen,  if  one  of  the 
lenses  1,  2,  or  3  is  inserted  in  the  fork,  as  the  distance  of  the  image 
is  then  diminished,  so  that  the  rays  diverge,  instead  of  emerging 
parallel. 

This  method  has  the  objection  of  showing  only  a  small  portion 
of  the  retina  at  a  time,  and  of  bringing  the  observer  too  near  the 
eye  for  convenience.  Another  method  is  therefore  more  com- 
monly used,  in  which  an  aerial  image  is  formed  in  front  of  the 
eye,  by  a  convex  lens,  and  this  is  viewed  either  directly  with  the 
eye,  or  with  a  second  convex  lens  of  long  focus. 

Hold  the  large  lens,  or  objective,  two  or  three  inches  in  front  of 
the  model,  with  the  left  hand,  steadying  it  with  the  little  finger, 
which,  in  the  case  of  the  real  eye,  rests  on  the  forehead  of  the 
patient.  The  mirror  should  be  held  a  foot  or  more  distant,  and 
turned  into  such  a  position  as  to  reflect  the  light  of  the  gas  flame 
into  the  model.  After  a  few  trials  a  very  beautiful  view  of  the 
retina  will  be  obtained.  The  image  will  be  inverted,  and  may  be 
made  as  large  as  the  objective  by'  removing  the  latter  to  a  distance 
equal  to  its  focal  length.  To  view  the  other  portions  of  the  retina, 
the  model  must  be  turned  from  side  to  side,  or  the  patient  re- 
quested to  direct  his  eye  towards  various  points  in  turn.  The 
image  is  improved  by  placing  lens  4  or  5  behind  the  mirror.  In  all 
these  experiments,  if  near-sighted,  use  a  lens  with  a  number  lower 
than  that  here  recommended;  thus,  instead  of  2,  use  1,  for  5,  use 
4,  etc.  The  first  of  the  above  methods,  that  is,  without  the  ob- 
jective, is  called  the  direct,  the  second  the  indirect  method. 

Now  screw  the  larger  diaphragm  over  the  lens,  and  try  once 
more  to  view  the  image ;  then  replace  it  by  the  small  diaphragm, 
with  which  it  is  about  as  diificult  to  observe  the  retina  as  with  the 
eye  in  its  normal  condition.  The  larger  diaphragm  corresponds  to 
the  case  where  the  pupil  is  expanded  by  belladonna.  With  the 
small  diaphragm  it  is  easier  to  look  a  little  obliquely  into  the  eye, 
thus  avoiding  the  light  reflected  from  the  lens,  which  gives  bright 
reflected  images  of  the  mirror. 

When  the  eye  is  near-sighted,  or  myopic,  the  retina  is  beyond 
the  principal  focus  of  the  lens.  This  effect  is  produced  in  the 
model  by  partly  unscrewing  the  lens,  taking  care  that  it  does  not 
fall  out.  An  image  of  the  retina  is  thus  formed  in  front  of  the 


198  OPTHALMOSCOPE. 

lens,  or  the  rays  from  it  converge.  Hence  when  employing  the 
direct  method,  a  concave  lens,  as  1  or  2,  must  be  placed  behind  the 
mirror,  to  render  the  image  visible.  When  the  eye  is  far-sighted,  or 
hypermetropic,  and  incapable  of  viewing  near  objects,  the  retina  is 
nearer  the  lens  than  its  focus.  Hence  the  image  is  formed  at  a 
considerable  distance  behind  the  lens,  and  can  readily  be  viewed 
by  the  direct  method  without  any  lens.  This  effect  is  imitated  by 
using  the  lens  marked  H,  which  has  a  longer  focus  than  the  other. 
The  difference  is  not  perceptible  by  the  indirect  method,  since  it  is 
neutralized  by  a  slight  motion  of  the  mirror. 

Sometimes  the  cornea  has  a  different  curvature  in  horizontal  and 
vertical  planes ;  it  is  then  said  to  be  astigmatic.  The  third  lens 
marked  A,  shows  this  defect.  With  this,  it  is  a  little  difficult  to 
view  the  retina  clearly,  since  the  focus  is  different  in  different 
planes ;  it  will  be  noticed,  however,  that  the  papilla  assumes  an 
elliptical,  instead  of  a  circular  form. 

Now  replace  the  emmetropic  lens,  and  insert  the  various  diseased 
retinas  in  turn,  using  the  small  diaphragm,  or,  if  necessary,  the 
larger  one.  The  retinas  are  numbered,  and  represent  the  following 
conditions  of  the  eye  :  — 

1.  Normal  retina. 

2.  Atrophy  of  the  papilla  and  retina. 

3.  Atrophy  of  the  choroid. 

4.  Staphyloma  posterior,  an  old  case ;  blood  focus  near  the 
macula  lutea. 

5.  Hemorrhage  of  the  retina. 

6.  Alteration  of  the  retina. 

7.  Staphyloma  posterior.     Separation  of  the  retina. 

8.  Infiltration  of  the  papilla  with  blood. 

9.  Exudation  of  serous  fluid,  between  the  choroid  and  retina. 

10.  Glaucoma,  with  the  circle  of  atrophy  of  the  choroid  around 
the  papilla. 

11.  Glaucoma  and  hemorrhage  of  the  retina. 

12.  Atrophy  of  the  papilla,  and  of  the  choroid  around  it. 

The  papilla  is  the  point  of  entrance  of  the  optic  nerve.  The 
macula  lutea,  the  point  of  the  retina  most  used. 

Atrophy  means  the  gradual  wasting  away,  and  absorption  of 
any  substance.  Staphyloma,  a  thinning  of  the  covering  of  the 


INTERFERENCE    OF    LIGHT.  199 

eyeball,  especially  around  the  optic  nerve,  allowing  this  portion 
of  the  ball  to  extend  backwards.  Glaucoma  is  an  increase  in 
quantity  of  the  vitreous  humor  within  the  eye,  causing  a  disten- 
tion  of  the  eyeball,  accompanied  with  acute  pain. 

89.    INTERFERENCE  OF  LIGHT. 

Apparatus.  To  observe  the  interference  of  light,  a  diffraction 
bank  is  employed,  which  consists  of  a  long  horizontal  bar  divided 
into  millimetres,  and  carrying  sliding  uprights,  to  which  the  fol- 
lowing instruments  may  be  attached,  and  placed  at  any  desired 
distance  apart.  A  cylindrical  lens  to  produce  a  bright  line  of 
light,  and  a  brass  plate  with  a  slit  in  it  of  variable  width,  like 
that  of  a  spectroscope.  A  biprism,  or  prism  of  glass  with  a  very 
obtuse  angle,  by  means  of  which  two  closely  adjacent  images 
of  any  object  will  be  formed,  and  a  double  mirror  designed  for 
the  same  purpose,  whose  two  halves  are  inclined  at  a  very  small 
angle,  which  may  be  varied  by  means  of  adjusting  screws.  To 
observe  the  various  effects  produced,  one  of  the  uprights  carries 
a  spider-line  micrometer,  or  a  simple  eye-piece  with  cross  hairs, 
which  may  be  moved  laterally,  and  its  position  determined  by  a 
millimetre  scale  and  index.  Or,  this  may  be  replaced  by  a  small 
direct  vision  spectroscope,  to  analyze  any  portion  of  the  light 
passing  through  the  instrument.  A  screen  of  ground  glass,  or 
paper,  may  also  be  substituted  for  the  eye-piece.  Although  some 
of  the  simpler  phenomena  are  visible  by  ordinary  light,  yet  to 
obtain  the  best  results  sunlight  is  indispensible.  An  arrangement 
is  also  desirable  by  which  an  intense  monochromatic  light  may  be 
obtained,  which  may  be  done  roughly  by  interposing  colored 
glasses,  but  much  better  by  placing  a  prism  in  front  of  the  slit, 
throwing  a  ray  of  sunlight  through  it,  and  projecting  the  spectrum 
thus  obtained  on  the  slit.  When  the  day  is  cloudy,  a  soda  or 
lithium  name  may  be  employed.  A  cover  should  be  placed 'over 
the  whole  to  cut  off  the  stray  light,  or  a  simple  piece  of  black 
cloth  may  be  employed  for  the  same  purpose. 

Experiment.  According  to  the  Undulatory  Theory  all  space  is 
supposed  to  be  filled  with  a  very  rare  medium,  called  ether,  whose, 
vibrations  give  rise  to  the  phenomena  of  light.  A  luminous  point 
throws  out  concentric  spherical  waves,  whose  diameters  increase 
with  very  great  velocity,  and  each  of  whose  radii  is  called  a  ray 
of  light.  The  direction  of  the  vibrations  is  transverse,  that  is,  per- 
pendicular to  the  ray,  as  is  the  case  of  waves  of  water,  and  the 
terms  crest  and  trough  are  here  also  used  to  denote  the  two 
opposite  positions  of  any  portion  of  the  ether.  The  distance  from 


200  INTERFERENCE    OF    LIGHT. 

one  crest  to  the  next  determines  the  color  of  the  ray,  and  is  called  the 
wave-length.  In  the  same  way,  the  intensity  of  the  light  depends 
the  height  of  the  wave,  or  distance  traversed  by  each  particle.  A 
particle  of  ether  can  receive  any  number  of  systems  of  vibrations, 
whatever  their  wave-length,  intensity,  direction,  or  plane  of  vibra- 
tion, and  will  transmit  each  precisely  as  if  the  others  did  not  exist. 

If  a  particle  receives  two  rays  of  light,  precisely  similar  in  every 
respect,  under  the  influence  of  both,  its  motion  will  be  increased, 
and  a  more  intense  light  produced.  Now  suppose  one  of  the  rays 
is  retarded  by  half  a  wave-length ;  its  crests  will  coincide  with,  and 
neutralize  the  troughs  of  the  other  ray ;  accordingly  the  particle 
will  not  move  at  all,  and  the  result  will  be  darkness.  The  same 
effect  will  also  be  produced  if  one  ray  is  retarded  three,  five,  or  in 
fact  any  odd  number  of  half  wave-lengths,  while  if  the  retardation 
is  an  even  number  of  half  wave-lengths,  crest  will  fall  upon  crest, 
and  the  light  will  be  increased.  This  neutralization  of  one  ray 
by  another,  or  light  added  to  light,  producing  darkness,  is  called 
interference,  and  by  means  of  it  many  most  important  laws  have 
been  established. 

To  produce  interference,  two  precisely  similar  sources  of  light 
are  needed,  at  a  very  short  distance  apart.  For  this  purpose, 
place  the  cylindrical  lens  on  a  support  at  one  end  of  the  diffraction 
bank,  and  throw  a  beam  of  sunlight  through  it.  A  very  narrow 
line  of  light  is  thus  produced  at  its  focus.  Place  the  biprism  on  a 
support,  at  a  short  distance  in  front  of  it,  and  two  images  will  be 
formed  very  near  together,  and  precisely  alike.  If,  now,  a  screen 
is  placed  near  the  other  end  of  the  bank,  its  centre  will  appear 
bright,  since  being  equidistant  from  both  images  it  will  receive 
simultaneously  the  crests  and  troughs  of  them  both.  If,  how- 
ever, a  point  is  taken  on  one  side  of  the  centre,  it  will  be  nearer 
one  image  than  the  other,  and,  accordingly,  the  crests  and  troughs 
will  not  arrive  simultaneously.  If,  then,  the  difference  of  path 
is  an  odd  number  of  half  wave-lengths,  darkness  will  be  produced, 
while  an  even  number  will  give  brightness. 

The  consequence  will  be  a  series  of  vertical  black  bands,  corres- 
ponding to  1,  3,  5,  etc.,  half  wave-lengths.  As,  however,  the  light 
is  white,  and  is  composed  of  rays  of  all  colors,  and  various  wave- 
lengths, the  bright  and  dark  spaces  will  be  at  different  distances 


INTERFERENCE    OF    LIGHT.  201 

for  each,  consequently  a  series  of  colored  bands  will  be  produced, 
with  a  white  centre.  These  bands  are  much  more  visible  with  the 
eye-piece,  and  their  number  is  increased  by  employing  monochro- 
matic lights. 

Their  position  affords  a  means  of  determining  approximately,  the 
length  of  a  wave  of  light,  as  follows.  Bring  the  cross-hairs  to 
coincide  successively,  with  each  of  the  visible  bright  bands,  when 
monochromatic  light  is  used,  and  read  the  position  of  the  eye- 
piece. Measure,  also,  the  distance  from  the  slit,  or  focus  of  the 
cylindrical  lens  to  the  cross-hair.  If,  now,  the  distance  between 
the  two  images  can  be  determined,  a  simple  calculation  will  give 
the  difference  in  their  distances  from  the  bright  band,  that  is,  one, 
two  or  three  times  the  wave-length,  according  to  the  number  of 
the  band.  The  distance  between  the  images  may  be  found  by  the 
following  device.  Insert  a  lens  between  them  and  the  eye-piece, 
having  a  focal  length  about  one  fourth  their  distance  apart. 
There  will  be  two  positions,  in  which  the  images  will  be  distinctly 
seen.  Bring  the  cross  hairs  to  coincide  with  their  images,  as 
formed  with  the  lens,  by  moving  it  laterally,  taking  care  to  move 
the  lens  only,  when  focussing.  In  one  case,  the  distance  between 
the  two  luminous  lines  will  be  magnified,  and  in  the  other,  dimin- 
ished, in  precisely  the  same  ratio.  Accordingly,  the  mean  pro- 
portional of  these  two  measurements  will  give  the  true  distance 
with  accuracy. 

To  calculate  the  wave-lengths  from  these  measurements,  let  D 
be  the  distance  from  the  slit  to  the  cross  hairs,  d  the  distance 
apart  of  the  two  images,  and  b  the  distance  of  the  first  band  from 
the  centre,  equals  one  half  the  distance  between  the  images  on  op- 
posite sides  of  the  centre  line.  Then  by  similar  triangles,  since 
b  and  d  are  always  very  small,  compared  with  D,  it  follows  that 
D  :  ~b  =  d:  l  the  required  wave-length.  In  the  same  way  the 
other  bands  give  2x,  3A,  etc.  Repeat  this  measurement,  with 
other  positions  of  the  eye-piece,  and  rays  of  other  colors,  and 
notice  that  the  more  refrangible  the  ray,  the  shorter  the  wave- 
length. 

As  the  position  of  the  eye-piece  is  varied,  any  given  band  will 
evidently  lie  on  a  hyperbola  with  the  two  images  as  foci,  since  it 
is  the  locus  of  a  point,  whose  distances  from  two  others  differs  by 


202  DIFFRACTION. 

a  constant  amount.  To  prove  this,  observe  the  position  of  one  of 
the  outer  bands,  varying  D  five  centimetres  at  a  time,  and  repre- 
sent the  results  by  a  curve,  in  which  D  gives  the  abscissas,  and 
an  enlarged  value  of  5,  the  ordinates.  Construct  also  the  theoret- 
ical curve  on  the  same  sheet. 

Now  replace  the  eye-piece  by  the  spectroscope,  and  moving  it 
laterally,  the  bright  and  dark  bands,  in  turn,  fall  on  the  slit,  and 
the  colors  of  which  they  are  composed  may  then  be  determined 
with  precision.  Thus  starting  with  the  central  bright  band,  it  will 
be  found  to  give  a  continuous  spectrum,  traversed,  of  course,  by 
the  usual  solar  lines.  In  the  first  black  band,  all  the  colors  disap- 
pear, then  all  the  colors  reappear,  beginning  with  the  violet,  and 
as  the  slit  is  moved  still  further,  a  dark  band  will  enter  the  violet 
end  of  the  spectrum,  and  will  traverse  it,  soon  to  be  followed  by  a 
succession  of  others,  whose  distance  apart  becomes  less  and  less, 
until  finally,  many  are  visible  in  the  spectrum  at  the  same  time. 

Similar  effects  may  also  be  obtained  with  the  double  mirror, 
taking  care  that  the  two  halves  are  slightly  inclined,  and  that 
their  edges  meet  exactly.  This  is  accomplished  by  means  of  the 
adjusting  screws.  It  will  then  form  two  closely  adjacent  images 
of  any  object  reflected  in  it.  Place  the  mirror  in  such  a  position, 
that  it  shall  reflect  two  images  of  the  slit  (which  is  moved  a 
short  distance  to  one  side)  along  the  bank,  when  the  bands  are 
formed  by  interference,  precisely  as  with  the  biprism.  A  dia- 
phragm must  be  interposed,  to  cut  off  the  direct  rays  of  the  slit 
from  the  eye-piece.  As  the  interval  between  the  two  images 
diminishes,  the  bands  become  more  spread  out,  as  may  be  shown 
by  diminishing  the  inclination  of  the  two  halves  of  the  mirror, 
by  means  of  the  adjusting  screws. 

90.    DIFFRACTION. 

Apparatus.  Besides  the  diffraction  bank  employed  in  Experi- 
ment 89,  a  number  of  brass  plates  are  needed,  which  may  be 
inserted  in  the  sliding  uprights,  and  which  are  perforated  with 
apertures  of  various  shapes  and  sizes.  Thus  one  will  carry  a  slit 
of  adjustable  width,  a  second  a  large  aperture  half  covered  by  a 
plate  with  a  vertical  edge,  others  with  two  closely  adjacent  slits, 
a  vertical  wire,  circular  holes  of  various  sizes,  and  some  with  two 
holes  a  short  distance  apart.  An  immense  variety  of  effects  may 
be  obtained  by  using  apertures  of  different  shapes,  and  sometimes 


DIFFRACTION.  203 

a  number  of  these  are  photographed  on  a  plate  of  glass  and 
brought  successively  into  the  axis  of  the  instrument.  A  plate 
of  glass  on  which  a  large  number  of  equidistant  fine  lines  are 
ruled  is  also  needed,  and  pieces  of  wire  gauze  and  lace  with 
square  and  hexagonal  meshes.  The  effect  of  these  various  ap- 
ertures is  best  shown  by  placing  them  in  front  of  a  telescope 
directed  toward  an  artificial  star,  as  in  Experiment  84,  or  the 
optical  circle  may  be  used,  replacing  the  slit  by  a  minute  circular 
aperture. 

Experiment.  The  phenomena  of  diffraction  are  best  explained 
by  a  comparison  with  the  similar  effects  produced  by  water. 
Suppose  a  long  straight  wave,  like  a  breaker  rolling  in  on  a 
beach,  encounters  in  its  passage  a  rock,  or  the  end  of  a  break- 
water. After  passing,  instead  of  leaving  the  water  quite  at  rest 
behind  the  obstacle,  the  end  of  the  wave  will  spread  inwards,  so 
that  if  the  rock  is  small  the  two  portions  will  meet  after  moving 
some  distance,  and  no  portion  of  the  surface  not  close  to  the  rock 
will  remain  perfectly  level.  When  the  undulatory  theory  was 
first  proposed,  it  was  claimed  by  its  opponents  that  the  waves  of 
light  would,  in  like  manner,  pass  around  an  obstacle,  so  that 
shadows  could  not  exist,  and  again  that  a  beam  of  light  could 
have  no  definite  edges,  since  it  would  spread  out  on  all  sides  like  a 
wave  of  water  after  entering  the  narrow  inlet  to  a  bay.  In  point 
of  fact,  this  spreading  actually  takes  place,  but  as  each  wave  is 
followed  by  millions  of  others  similar  to  it,  interference  takes  place 
so  that  but  little  remains,  except  in  the  direction  of  the  beam. 
In  fact,  it  is  only  by  taking  special  precautions  that  the  remaining 
rays  can  be  detected,  and  the  phenomena  then  observed  are 
known  by  the  name  of  diffraction. 

Place  the  cylindrical  lens,  or  the  slit,  at  one  end  of  the  diffraction 
bank,  and  throw  a  beam  of  sunlight  through  it.  Place  a  screen  at 
the  other  end,  and  between  them  the  plate  with  the  aperture  half 
closed  by  a  piece  of  sheet  brass  having  a  vertical  edge,  or  a  slit 
with  one  side  removed.  The  shadow  will  now  be  cast  upon  the 
latter,  and  examining  its  edge  closely,  it  will  be  found  that  a  small 
amount  of  light  has  been  bent  inwards  or  inflected.  Outside  of 
the  edge,  and  parallel  to  it,  a  series  of  colored  bands  or  diffraction 
fringes  appear,  due  to  the  partial  interference  of  these  rays.  They 
are  seen  on  a  much  larger  scale  by  means  of  the  eye-piece,  and 


204 


DIFFRACTION. 


their  constitution  is  well  shown  by  means  of  the  direct-vision 
spectroscope,  as  is  the  case  of  the  interference  fringes. 

Now  replace  the  other  side  of  the  slit,  and  as  it  is  gradually 
narrowed  new  fringes  will  appear  in  the  shadow,  which  finally  will 
quite  obscure  the  others,  leaving  an  appearance  very  like  that  pro- 
duced by  interference  with  a  biprism.  When  the  slit  is  moder- 
ately narrow,  both  system  of  fringes  are  visible,  those  in  the 
interior  being  either,  bright  or  dark-centered,  according  to  the  dis- 
tance of  the  screen.  Next  use  a  narrow  screen,  as  a  wire,  instead 
of  the  slit,  in  the  middle  of  the  bank,  when  a  series  of  fringes  will 
be  obtained,  both  inside  and  outside  the  shadow,  and  varying 
their  distance  apart  with  the  diameter  of  the  wire.  By  using  two 
slits  near  together,  fringes  are  also  produced  by  interference  as 
with  the  biprism.  Strangely  enough,  these  efforts  are  quite  in- 
dependent of  the  material  of  the  slit,  its  thickness,  or  physical  state. 
After  verifying  the  above  facts,  measure  the  form  of  some  of 
the  fringes,  as  in  the  case  of  interference,  and  see  if  their  relative 
distances  from  the  centre  agree  with  theory. 

When  rays  from  a  real  or  artificial  star  pass  through  an 
aperture,  those  striking  the  edges  are  diffracted,  so  that  they 
are  thrown  off  obliquely  in  directions  dependent  on  their  wave- 
lengths and  the  form  of  the  aperture.  I£  therefore,  they  are 
received  in  a  telescope,  the  direct  rays  will  form  a  bright  spot, 
or  image  of  the  star,  which  will  be  surrounded  with  colored 
fringes  or  bands  of  very  various  forms. 

To  begin  with  the  simplest  case,  suppose  the  aperture  circular 
and  of  considerable  size,  as  in  a  common  telescope.  The  dif- 
fraction will  be  very  slight,  but  quite  perceptible,  with  a  good 
instrument  and  a  high  power,  although  with  a  poor  lens  it  is  some- 
times obscured  by  the  aberration.  The  true  angular  diameter 
of  the  fixed  stars  is  so  exceedingly  small  that  it  would  be  quite  im- 
possible to  observe  their  disks  with  any  power  yet  employed. 
With  a  good  telescope,  however,  small  bright  circles  are  seen, 
called  spurious  disks,  which  increase  in  diameter  as  the  aperture 
is  diminished,  and  which  are  surrounded  by  one  or  more  colored 
rings,  due  also  to  diffraction.  If,  now,  a  triangular  aperture  is 
interposed  in  front  of  the  objective,  the  star  is  seen  to  be  sur- 
rounded with  six  diverging  rays,  corresponding  to  the  angles  and 


WAVE    LENGTHS.  205 

centres  of  the  sides  of  the  triangle,  while  a  square  aperture  gives 
a  star  with  four  rays.  Try,  in  the  same  way,  the  other  apertures 
and  the  gauze,  with  which  a  vast  variety  of  curious  effects  may  be 
obtained.  On  interposing  the  series  of  equidistant  lines,  a  number 
of  colored  bands  are  seen  on  each  side  of  the  star,  with  their 
violet  ends  towards  the  centre,  and  their  direction  perpendicular 
to  the  lines  on  the  glass.  This  effect  is  most  important,  as  it 
affords  a  means  of  determining  the  length  of  a  ray  of  light  with 
the  utmost  precision,  as  will  be  described  in  the  next  experiment. 

In  all  cases  the  distances  of  the  colored  rays  will  depend  on 
their  wave-lengths,  being  greatest  for  the  red  and  yellow,  then  for 
green,  and  least  for  the  blue  and  violet.  This  may  be  shown  by 
employing  monochromatic  light,  which  may  be  obtained  by  illumi- 
nating the  slit  or  artificial  star  by  different  parts  of  the  spectrum, 
formed  by  allowing  a  ray  of  sunlight  to  pass  through  a  prism,  or 
more  simply  by  interposing  colored  glass,  or  employing  a  soda 
flame. 

Many  familiar  phenomena  are  due  to  d infraction ;  for  example, 
the  halo  around  a  distant  light  seen  through  a  window  covered 
with  moisture  or  frost,  or  the  same  effect  produced  on  the -sun  or 
moon  by  fog.  When  the  particles  are  all  of  nearly  the  same  size, 
colors  become  visible,  which  may  be  distinguished  from  the  ordi- 
nary large  halos,  both  by  their  small  size,  and  by  noticing  that  in 
halos  produced  by  diffraction  the  red  is  always  outside,  while  in 
halos  caused  by  refraction  in  minute  crystals  of  ice,  the  red  is 
always  inside,  since  this  color  has  a  smaller  index  of  refraction, 
but  greater  wave-length.  This  effect  may  be  produced  artifi- 
cially by  scattering  on  a  plate  of  glass,  lycopodium,  or  any  fine 
powder  whose  particles  are  all  of  nearly  the  same  size. 

91.     WAVE    LENGTHS. 

Apparatus.  The  optical  circle,  and  a  glass  plate,  on  which  are 
ruled  several  thousand,  very  fine,  equidistant  lines,  at  intervals  of 
about  one  hundredth  of  a  millimetre.  Sunlight  is  desirable  for 
this  Experiment,  but  if  the  day  is  cloudy,  a  soda  and  lithium 
flame  may  be  employed  instead. 

Experiment.  One  of  the  most  important  applications  of  diffrac- 
tion, is  to  the  measurement  of  the  lengths  of  waves  of  light  of 


206  WAVE    LENGTHS. 

various  colors.  The  fringes  obtained  with  screens  traversed  by 
very  fine  lines,  or  gratings,  as  they  are  called,  are  employed  for  this 
purpose,  and  give  very  accurate  results. 

Sejt  the  two  telescopes  of  the  optical  circle  opposite  each  other, 
adjust  them  for  parallel  rays,  and  place  the  glass  plate  on  the 
centre  stand  between  them,  and  at  right  angles  to  their  axes. 
Reflect  a  ray  of  sunlight  through  the  slit  of  the  collimator  by 
means  of  the  mirror,  and  on  looking  through  the  observing  tele- 
scope, the  following  appearances  will  be  visible  as  it  is  moved  from 
side  to  side.  In  the  centre  will  be  a  brilliant  white  image  of  the 
slit,  and  on  each  side  spectra  will  be  seen,  with  their  violet 
ends  turned  towards  it.  The  first  will  be  bright  and  short, 
and  each  in  turn  fainter  and  longer,  until  finally  they  over- 
lap, forming  a  continuous  band  of  light.  Focus  with  care,  and 
nearly  close  the  slit,  when  the  solar  lines  will  become  visible  in 
each  spectrum,  as  in  a  spectroscope,  except  that  in  the  present 
case  the  red  end  is  much  more  extended,  the  violet  more  crowded 
together.  Care  must  be  taken  that  the  lines  are  vertical,  as  the 
spectra  being  perpendicular  to  them  will  otherwise  fall  out  of  the 
field,  above  on  one  side,  and  below  on  the  other.  The  plate  should 
also  be  perpendicular  to  the  axis  of  the  collimator,  or  an  error 
will  be  introduced  in  the  angular  distance  of  the  spectra. 

The  formation  of  these  spectra,  is  explained  as  follows.  Let  M^ 
-ZVJ  0,  Fig.  64,  represent  the  rays  from  the  slit,  after  being  rendered 

parallel  by  the  lens  of  the  collima- 
tor,  so  that  they  shall  all  be  in  the 
same  phase,  or  state  of  vibration, 
when  they  strike  the  plate  of  glass. 
The  lines  on  the  latter,  are  shown 
on  a  greatly  enlarged  scale,  at  A, 
13)  C,  J),  which  represent  a  section 
at    right    angles  to   their    length. 
Being    opaque,    or  at   least   only 
translucent,   they   divide  the  sur- 
face of  the  glass,  into  a  number  of  equidistant  narrow   apertures, 
through  which  the  light  passes.    As  was  proved  by  Huyghens, 
each  of  these  may  be  regarded  as  a  new  source  of  light,  from 


WAVE    LENGTHS.  207 

which  the  rays  pass  out  in  all  directions,  and  in  general,  interfere 
and  neutralize  each  other.  There  are,  however,  certain  directions, 
aaAP,m  which  the  difference  of  path  of  AP  and  BQ,  equals 
exactly  one  wave-length  /*,  and  in  this  case  they  will  unite, 
and  light  will  be  produced.  The  ray  CR  will  add  its  effect, 
since  it  differs  by  exactly  two  wave-lengths,  and  in  the  same 
way,  all  the  other  rays  from  the  other  apertures  unite  and  pro- 
duce a  bright  light  in  the  direction  AP.  The  direction  of  PR 
is  that  of  the  front  of  the  wave,  or  perpendicular  to  AP,  and 
drawing  AF  parallel  to  it,  the  condition  that  light  may  be  pro- 
duced is  evidently  that  in  the  right-angled  triangle  AFB,  FJB 
shall  equal  L  Call  d  the  distance  between  the  lines,  in  frac- 
tions of  a  millimetre,  and  a  the  angle  the  ray  AP  makes  with  the 
axis  of  the  collimator,  equals  FAS.  Then  X  =  d  sin  a,  from 
which,  knowing  d  and  a,  A  may  be  computed.  To  find  a,  bring 
the  cross-hairs  of  the  observing  telescope  successively  to  coincide 
with  the  image  of  the  slit  and  the  given  ray  in  the  first  spectrum, 
and  the  difference  in  the  readings  of  the  vernier  gives  the  required 
angle;  or  better,  read  the  position  of  the  correspondh.g  rays  in 
the  spectra  on  the  right  and  left  of  the  central  image,  and  divide 
the  difference  by  two.  Make  a  similar  observation  with  three  or 
four  of  the  prominent  lines.  Suppose,  now,  that  a  is  so  much  i 
increased  that  AF  =  2A ;  evidently  light  is  again  produced, 
which  gives  rise  to  the  second  spectrum  on  each  side.  The  third 
and  fourth  spectra  are  accounted  for  in  the  same  way.  Measure 
the  position  of  the  lines  before  observed  in  all  of  them,  and  com- 
pute X  from  each,  taking  care  to  divide  by  two  for  the  second,  by 
three  for  the  third,  etc.,  to  get  the  true  wave-length.  The  mean 
of  these  observations  should  then  agree  closely  with  that  given  on 
page  152.  If  d  is  not  given,  it  should  be  determined  on  the 
dividing  engine,  or  with  the  microscope  and  spider-line  microm- 
eter. 

If  the  lines  on  the  glass  plate  are  well  ruled,  very  beautiful 
spectra  may  be  obtained,  in  some  cases  almost  equal  to  those 
formed  by  the  best  spectroscopes.  Generally  the  spectra  on  one 
side  are  better  than  those  on  the  other,  probably  owing  to  some 
want  of  symmetry  in  the  two  sides  of  the  lines.  Again,  often 
one  of  the  spectra  will  be  fainter  than  the  next  beyond  it,  or  even 


208  POLARIZED    LIGHT. 

wanting  altogether,  it  may  be  proved  analytically  that  the  mih 
spectrum  will  be  wanting,  when  the  ratio  of  the  width  of  the 
lines  to  the  spaces  between  them,  is  as  n  :  n',  and  m  =  n  -f-  n'. 
That  is,  if  the  dark  spaces  are  half  as  broad  as  the  bright,  n  —  1, 
n'  =  2,  m  =  3,  and  the  third  spectrum  will  be  wanting. 

92.    POLARIZED  LIGHT. 

Apparatus.  A  rhomb  of  Iceland  spar,  and  examples  of  the  five 
methods  of  polarizing  light,  that  is,  by  reflection,  by  refraction  or 
by  a  bundle  of  plates  set  at  an  angle  of  55°,  and  by  the  three 
methods  of  double  refraction.  These  consist  of  a  double-image 
prism,  a  Nicol's  prism,  and  a  tourmaline  plate.  Fig.  65  represents 
a  form  of  polariscope  which  will  be  found  both  simple  and  effective. 
jB  is  a  plate  of  glass  resting  on  a  piece  of  black  velvet,  A 
a  screen  of  ground  glass,  and  D  a  Nicol's  prism.  This  is  so 
placed  that  the  angle  of  incidence  of  the  ray  reflected  from  the 
centre  of  B  shall  be  55°,  and  consequently  shall  be  totally  polar- 
ized. C  is  a  plate  of  glass  on  which  the  object  to  be  examined  is 
laid.  Various  objects  should  accompany  this  instrument,  as  a 
plate  of  selenite,  some  figures  made  of  the  same  material,  some 
pieces  of  uriannealed  glass,  and  two  small  screw-presses  by  which 
small  squares  or  rods  of  glass  may  be  subjected  to  longitudinal 
or  transverse  strain.  Also  some  lenses,  glass  stoppers,  and  other 
articles  imperfectly  annealed,  and  some  spectacle  lenses  of  quartz 
and  glass.  Plates  of  the  following  series  of  crystals  should  be 
provided.  1.  Iceland  spar  cut  perpendicular  to  the  axis;  2, 
quartz ;  3,  arragonite ;  4,  topaz ;  5,  borax ;  6,  nitre  ;  7,  double  plate 
of  quartz  giving  hyperbolas ;  8,  Savart's  bands.  This  list  may  be 
greatly  extended,  and  it  is  well  to  add  any  novelties  that  can  be 
found,  with  a  written  description  appended.  All  these  objects 
will  appear  to  much  greater  advantage  if  the  outside  light  is  cut 
off,  either  by  a  black  cloth,  or  by  a  cover  fitting  over  the  polari- 
scope, and  extending  from  A  to  D  in  the  figure. 

Experiment.  According  to  the  Undulatory  Theory,  light  is 
produced  by  vibrations  of  the  ether  at  right  angles  to  the  direc- 
tion of  the  ray.  If,  then,  the  latter  moves  vertically,  all  the  mo- 
tions will  be  horizontal,  and  in  common  light,  some  north  and 
south,  others  east  and  west,  and  others  in  various  intermediate 
directions,  that  is,  in  all  planes  passing  through  the  ray.  If,  now, 
the  vibrations  can  in  any  way  be  confined  to  a  single  plane,  the 
light  is  said  to  be .  polarized,  and  this  plane  is  called  the  plane  of 
vibration  of  the  ray.  A  plane  perpendicular  to  this  and  passing 


POLARIZED    LIGHT.  209 

through  the  ray,  is  called  the  plane  of  polarization.  It  would 
be  much  better  to  have  taken  the  latter  plane  as  coincident  with 
the  former,  but,  unfortunately,  the  name  was  given  before  the 
direction  of  the  vibrations  was  known. 

Although  it  is  impossible  to  detect  by  the  eye  alone  whether 
light  is  polarized  or  not,  yet  many  substances  affect  it  differently, 
according  to  the  direction  its  plane  bears  to  some  line  in  them,  so 
that  when  it  emerges  from  them,  it  no  longer  possesses  the  prop- 
erties of  plane  polarized  light.  To  examine  the  effect  produced, 
the  ray  is  first  passed  through  the  polarizer,  as  it  is  called,  by  which 
all  its  vibrations  are  brought  into  one  plane,  it  is  then  allowed  to 
pass  through  the  substance  under  examination,  and  finally  tested 
by  the  analyzer,  which  may  be  made  precisely  like  the  polarizer, 
and  is  used  to  detect  any  change  effected  in  the  ray. 

There  are  five  forms  of  polarizers  in  common  use.  First,  by  re- 
flection. When  a  ray  of  polarized  light  impinges  on  a  plane 
surface  of  a  transparent  medium,  the  amount  reflected  depends  on 
the  direction  of  the  plane  of  polarization,  and  the  angle  of  inci- 
dence. If  the  plane  is  perpendicular  to  the  plane  of  incidence, 
and  the  angle  is  such  that  the  reflected  and  refracted  rays  shall  be 
perpendicular,  all  the  light  is  transmitted.  The  angle  of  incidence 
is  then  called  the  angle  of  total  polarization,  and  its  value  may  be 
determined  as  follows.  Let  n  be  the  index  of  refraction  of  the 
medium,  i  the  angle  of  incidence,  which  equals  the  angle  of  reflec- 
tion, and  r  the  angle  of  refraction.  Then,  since  the  reflected  and 
refracted  rays  are  at  right  angles,  *  +  r  —  90°,  but  sin  i  =  n  sin 
r  =  n  sin  (90  -  i)  =  n  cos  r,  and  tang  i  =  n.  Common  light 
being  composed  of  rays  polarized  in  every  plane  passing  through 
the  beam,  may  be  regarded  as  composed  of  two  equal  rays,  polar- 
ized at  right  angles,  just  as  all  forces  acting  on  a  point  in  a  plane 
may  be  divided  into  two  components  at  right  angles  to  each  other. 
Regarding,  then,  the  light  as  composed  of  two  beams,  one  A,  polar- 
ized in  the  plane  of  incidence,  and  the  other,  .2?,  polarized  at  right 
angles  to  it,  evidently  none  of  the  latter  will  be  reflected ;  hence 
the  reflected  ray  will  be  entirely  composed  of  light  polarized  in  the 
plane  of  incidence,  or  will  be  totally  polarized.  The  value  of 
*  is  about  55°  for  glass,  and  53°  for  water.  The  simplest  form  of 
polarizer  is  therefore  a  plate  of  glass,  on  which  the  light  impinges 

14 


210  POLARIZED    LIGHT. 

at  an  angle  of  55°.  Commonly  the  lower  surface  is  blackened,  or 
black  glass  is  employed,  but  there  is  no  advantage  in  this,  in  fact 
it  is  better  to  use  several  plates  of  clear  glass,  to  increase  the 
light. 

As  a  portion  of  A  is  turned  back  by  the  glass,  evidently  the 
refracted  beam  will  be  partially  polarized,  being  composed  of  the 
whole  of  -Z?,  and  part  of  A.  By  using  a  number  of  plates,  each 
will  reflect  a  portion  of  -4,  leaving  B  unaffected ;  the  latter  may 
thus  be  almost  completely  freed  from  A>  or  the  light  nearly 
perfectly  polarized. 

The  other  three  polarizers  depend  on  double  refraction.  If  a 
ray  of  light  is  allowed  to  pass  through  any  crystal  not  of  the 
monometric  system,  it  will  be  divided  into  two  parts,  one  called 
the  ordinary  ray,  which  will  follow  the  usual  laws  of  refraction, 
and  the  other,  the  extraordinary  ray,  which  will  follow  new  laws. 
To  show  this,  lay  a  crystal  of  Iceland  spar  on  a  piece  of  paper  on 
which  is  marked  a  single  dot.  The  latter  will  now  appear  double, 
and  if  the  crystal  is  turned,  one  image,  the  extraordinary,  will 
revolve  around  the  other.  These  two  rays  are  found  to  be  polar- 
ized in  planes  at  right  angles  to  each  other,  but  in  the  present 
case  are  not  sufficiently  separated  to  be  conveniently  employed. 
They,  moreover,  emerge  parallel,  that  is,  they  are  no  more  sep- 
arated for  distant  objects  than  for  near,  since  the  plate  being 
bounded  by  parallel  faces,  the  second  surface  neutralizes  the  angu- 
lar divergence  produced  by  the  first.  To  remedy  this  defect, 
prisms  of  glass  and  spar  are  cemented  together  in  such  a  way 
that  the  refraction  of  one  ray  shall  be  compensated,  while  the 
other  will  pass  out  obliquely,  giving  two  images  separated  by  an 
angular  amount  of  two  or  three  degrees.  This  combination  is 
called  a  double-image  prism. 

Another  arrangement  is  the  Nicol's  prism,  which  consists  of  a 
rhomb  of  Iceland  spar  cut  diagonally,  and  the  two  parts  cemented 
together  again  with  Canada  balsam.  This  substance  has  an  index 
of  refraction  greater  than  the  extraordinary,  but  less  than  the 
ordinary  ray  in  spar,  consequently  the  former  will  pass  through 
unchanged,  while  the  latter  being  totally  reflected  will  be  thrown 
out  on  one  side,  and  will  be  absorbed  by  the  black  paint  covering 
the  prism.  The  light  passing  through  will  therefore  be  polarized 


POLARIZED    LIGHT.  2ll 

in  a  plane  passing  through  the  ray  and  the  longer  diagonal  of  the 
rhombus  at  the  end  of  the  prism.  This  is  called  the  principal 
plane,  or  simply  the  plane,  of  the  prism. 

The  fifth  form  of  polarizer  is  a  plate  of  tourmaline,  cut  parallel 
to  the  axis,  which  posseses  the  curious  property  of  absorbing  the 
ordinary  ray,  so  that  the  emergent  light  is  polarized  in  a  plane 
parallel  to  its  axis,  or  greater  diameter. 

Either  of  these  instruments  may  be  employed  as  a  .polarizer, 
but  each  has  its  special  advantages  and  defects.  The  method  of 
reflection  is  the  simplest,  and  a  beam  of  any  size,  perfectly  polar- 
ized may  be  obtained  by  it,  but  there  is  much  loss  of  light  from 
the  transmitted  rays,  and  the  change  of  direction  is  often  an 
objection.  The  bundle  of  glass  plates  give  a  large  beam,  but  the 
polarization  is  not  very  perfect  unless  a  large  number  of  plates  is 
employed,  and  then  the  loss  by  absorption  is  considerable.  By 
the  other  methods  very  large  beams  cannot  be  obtained.  The 
double  image  prism  gives  excellent  results  when  the  presence  of 
the  second  beam  is  not  objectionable,  or  when,  as  sometimes 
happens,  it  can  be  thrown  out  to  one  side  of  the  apparatus.  The 
Nicol's  prism  is  more  employed  than  any  other  polarizer,  but  when 
of  large  size  it  is  very  expensive.  A  tourmaline  plate  is  also  good, 
but  if  very  thin  the  ordinary  ray  is  not  wholly  absorbed,  and  the 
polarization  is  not  complete ;  while  if  thick,  the  ray  is  strongly 
colored.  Colorless  tourmalines  exist,  but  unfortunately  are  not 
opaque  to  the  ordinary  ray,  and  hence  do  not  polarize  the  trans- 
mitted light.  Examples  of  these  various  polarizers  will  be  found 
on  the  table. 

When  a  ray  of  polarized  light  is  viewed  through  a  Nicol's  prism, 
or  other  polarizer,  the  amount  of  light  transmitted  varies  as  the 
prism  is  turned.  Thus  allow  a  ray  of  light  to  pass  through  a 
Nicol's  prism  with  its  principal  plane  vertical,  that  is,  so  that  the 
transmitted  light  shall  be  polarized  in  a  vertical  plane.  If  this 
beam  is  viewed  with  a  second  Nicol's  prism,  it  will  be  found  that 
as  the  latter  is  turned,  the  amount  of  light  transmitted  varies, 
being  greatest  when  its  plane  is  vertical,  and  nothing  or  all  the 
light  cut  off,  when  the  plane  is  horizontal.  This  evidently  follows, 
since  the  prism  then  transmits  only  light  polarized  vertically.  In 
intermediate  positions,  the  amount  of  light  is  determined  by 


212 


POLARIZED    LIGHT. 


decomposing  the  ray  into  two  at  right-angles,  as  in  the  case  of 
forces,  only  it  will  be  proportional  not  to  the  cosine,  but  to  the 
square  of  the  cosine  of  the  angles.  Try  the  other  polarizer,  in  the 
same  way,  and  it  will  be  found  in  all  cases,  that  when  their  planes 
are  parallel,  light  is  transmitted,  but  when  turned  at  right-angles, 
or  crossed,  as  it  is  called,  the  light  is  cut  off.  Accordingly,  to 
test  for  polarized  light,  view  the  beam  through  a  Nicol's  prism  or 
tourmaline,  which  is  then  called  an  analyzer.  If  there  is  no  change 
of  brightness  of  the  transmitted  ray,  the  light  is  unpolarized,  while 
if  in  certain  positions  all  the  light  is  cut  off,  the  polarization  is 
complete.  Next,  turning  the  analyzer,  find  the  position  in  which 
the  field  is  darkest,  when  its  plane  will  be  perpendicular  to  the 
plane  of  polarization. 

To  apply  this  to  some  familiar  objects,  examine  the  light  re- 
flected from  the  top  of  a  varnished  table,  and  it  will  be  found  to 
be  strongly  polarized  in  a  vertical  plane.  Moreover,  when  this 
light  is  cut  off,  the  color  of  the  wood  and  its  grain  is  much  better 
seen.  It  has  been  proposed  to  use  Nicol's  prisms  in  this  way  for 
viewing  oil  pantings,  thus  cutting  oif  the  troublesome  reflection. 
Sometimes  the  light  reflected  from  the  front  of  glass  cases  renders 
it  difficult  to  distinguish  objects  within  them.  A  Nicol's  prism  is 
then  often  very  serviceable.  Again,  it  has  been  proposed  to  use 
a  Nicol's  prism  to  cut  off  the  light  reflected  by  water,  to  render 
rocks  or  other  objects  beneath  its  surface  more  visible.  To  show 
this  effect,  place  a  coin  at  the  bottom  of  a  vessel  of  water,  or 
under  several  plates  of  glass,  and  allow  a  strong  light  to  fall  on  it. 
It  may  then  be  easily  seen  when  the  polarized  light  is  cut  off, 
although  otherwise  quite  invisible.  As  another  example,  view  the 
two  dots  seen  through  a  crystal  of  Iceland  spar,  and  they  will  be 
found  polarized  in  planes  at  right-angles.  If  the  two  images  are 
connected  by  a  line,  it  will  lie  in  the  plane  of  the  ordinary  image, 
or  fixed  dot,  around  which  the  other  appears  to  revolve. 

To  view  any  body  by  polarized  light,  the  instrument  represented 
in  Fig.  65,  will  be  found  both  simple  and  effective.  B  is  the 
polarizer,  consisting  of  one  or  more  plates  of  glass,  and  D  a  Nicol's 
prism,  serving  as  an  analyzer  which  may  be  turned  by  any  desired 
amount,  and  which  is  set  at  such  an  angle  that  the  light  reflected 
from  the  centre  of  B  shall  be  totally  polarized ;  C  is  a  plate  of 


POLARIZED    LIGHT.  213 

glass  on  which  the  object  to  be  examined  may  be  laid,  and  A  a  piece 
of  ground  glass,  to  cut  off  the  reflection  of  outside  objects,  and  to 
render  the  field  of  view  bright  and 
uniform.     The  light  reflected  from 
B  is  polarized  vertically ;   accord- 
ingly, when  D  has  its  plane  verti.- 
cal,  the  field  is  bright,  when  hori- 
zontal, the  field  is  dark. 

Suppose  now,  any  doubly  re- 
fracting medium  is  inserted  be- 
tween the  analyzer  and  polarizer; 
for  instance,  a  plate  of  selenite  Fig>  ^ 

laid  upon  C.     The  ray  on  entering 

the  selenite  is  divided  into  two,  the  ordinary  and  extraor- 
dinary, polarized  at  right-angles,  the  plane  of  the  ordinary 
passing  through  the  axis  of  the  crystal.  The  relative  intensities 
of  the  two  will  depend  on  the  position  of  this  axis  with  regard  to 
the  plane  of  polarization  of  the  ray,  and  may  always  be  obtained 
by  decomposing  the  latter  into  two  parts;  one  the  ordinary  ray, 
coinciding  with  the  axis,  the  other  at  right  angles  to  it.  If  the  axis 
is  perpendicular  to  the  plane  of  polarization,  evidently  all  the 
light  will  pass  into  the  extraordinary  ray,  while  if  they  coincide 
all  becomes  ordinary.  The  two  intensities  are  equal  when  the 
angle  between  the  axis  and  plane  is  45°.  Now  the  two  rays  travel 
through  the  crystal  with  unequal  velocities,  as  is  shown  by  their 
different  indices  of  refraction.  On  emerging,  one  ray  will  be 
behind  the  other  by  an  amount  dependent  on  the  thickness  of  the 
plate ;  for  instance,  one  half  wave-length  of  yellow  light.  The 
two  rays,  however,  cannot  interfere,  since  they  are  polarized,  and 
are  therefore  vibrating,  at  right-angles.  Therefore  the  crystal 
will  still  appear  colorless  to  the  eye.  If,  however,  it  is  viewed  with 
a  Nicol's  prism  with  its  plane  vertical,  the  two  rays  are  again 
decomposed,  the  horizontal  components  cut  off,  and  the  vertical 
portions  brought  together  so  that  they  can  interfere.  If,  then, 
white  light  is  employed,  which  is  composed  of  rays  of  all  colors, 
the  yellow  portion  will  be  stricken  out,  and  the  remainder  will  be 
of  the  complementary  color,  or  purple.  Now  turn  the  analyzer 
90°.  The  rays  before  cut  off  will  now  be  transmitted,  and  vice 


214  POLARIZED    LIGHT. 

versa,  accordingly  the  color  of  the  light  will  be  yellow.  As  the 
analyzer  is  turned  from  these  points  the  colors  become  fainter  and 
fainter,  until  at  the  45°  points  all  the  rays  are  equally  affected,  and 
the  light  becomes  white. 

In  the  same  way,  on  turning  the  selenite  plate,  the  two  compo- 
nents are  equal  only  when  the  a^ds  is  inclined  at  an  angle  of  45°, 
in  which  case  the  interference  is  complete.  In  other  positions  one 
component  is  larger  than  the  other,  the  interference  is  only  partial, 
and  the  colors  are  fainter,  a  part  of  the  light  being  white.  When 
the  angle  becomes  0°,  or  the  axis  lies  in  the  plane  of  polarization, 
all  of  the  light  passes  into  the  ordinary  ray,  none  into  the  extra- 
ordinary, and  consequently  there  is  no  interference,  and  the  light 
is  white.  '  In  the  same  way,  when  the  axis  is  perpendicular  to  the 
plane  of  polarization,  all  the  light  enters  the  extraordinary  ray, 
and  the  result  is  again  white  light.  Accordingly  as  the  selenite  is 
turned,  the  color  becomes  fainter  and  fainter,  and  disappears  at 
the  0°  and  90°  points,  but  on  passing  them  does  not  assume  its 
complementary  tint. 

By  varying  the  thickness  of  the  plate,  the  amount  of  retard- 
ation may  be  varied  at  will,  and  with  it  the  wave-length  of  the 
ray  stricken  out,  and  the  color  produced.  Figures  are  sometimes 
made  of  selenite  to  represent  birds  or  flowers,  each  portions  hav- 
ing such  a  thickness  that  when  viewed  by  polarized  light  they  will 
assume  their  proper  colors.  They  are  then  mounted  in  Canada 
balsam  between  two  plates  of  glass,  and  by  ordinary  light  being 
transparent  are  almost  invisible.  Placing  them  on  GY,  however, 
they  appear  in  gorgeous  colors,  which  disappear  as  the  analyzer  D 
is  turned,  and  again  reappear  in  complementary  tints,  as  the  rota- 
tion is  continued.  If,  however,  the  selenite  is  turned,  the  colors 
fade,  but  reappear  unchanged. 

All  transparent  bodies  will  produce  double  refraction,  and  affect 
polarized  light  when  subjected  to  unequal  strains  in  different 
directions.  In  the  cases  mentioned  above,  this  is  effected  by 
crystallization,  but  it  may  also  be  produced  mechanically.  To 
show  this,  place  a  square  of  glass  in  the  small  press,  and  lay  it  on 
G7,  turning  D  so  as  to  cut  off  the  light.  No  effect  is  now  pro- 
duced, but  as  soon  as  the  glass  is  compressed  by  turning  the  screw, 
bright  spaces  appear  at  the  points  where  the  pressure  is  greatest, 


POLARIZED    LIGHT.  215 

and  which  as  the  screw  is  turned,  increase  in  size,  and  finally 
become  colored.  Care  must  be  taken  not  to  exert  too  great  a 
pressure  or  the  glass  may  be  fractured.  Apply  in  the  same  way 
a  transverse  strain  to  a  rod  of  glass  with  the  other  press,  and 
sketch  the  appearance.  Still  more  care  is  needed  in  this  case  not 
to  break  the  glass. 

When  glass  is  cooled  suddenly  the  exterior  contracts,  and  when 
the  interior  cools,  the  whole  is  subjected  to  great  strain,  rendering 
it  very  brittle.  To  remedy  this,  glass  vessels  intended  for  com- 
mon use  are  annealed,  that  is,  heated  arid  allowed  to  cool  very 
slowly.  Place  the  pieces  of  unannealed  glass  on  (7,  and  very 
curious  and  beautiful  markings  will  appear,  which  vary  with  the 
form  of  the  specimen,  and  the  position  of  the  analyzer.  Common 
glass  objects,  as  stoppers  to  bottles,  may  be  tested  in  the  same  way, 
to  see  if  they  have  been  properly  annealed.  The  best  practical 
application  made  of  this  principle,  is  to  test  lenses,  as  already 
mentioned  in  Experiment  84. 

When  a  ray  of  light  passes  through  a  crystal,  not  of  the  mono- 
metric  system,  the  effect  produced  varies  with  the  direction.  In 
the  dimetric  and  hexagonal  systems,  when  the  ray  passes  through 
the  crystal  in  the  direction  of  its  principal  axis,  it  is  not  divided 
into  two,  or  more  properly,  the  two  follow  the  same  path  but  with 
different  velocities.  This  direction  is  called  the  optic  axis  of  the 
crystal,  and  such  crystals  are  called  uniaxial.  Now  place  a  rhomb 
of  Iceland  spar  with  its  principal  axis  vertical,  that  is,  so  that  the 
corner  formed  by  the  angles  of  120°  shall  be  uppermost,  and  the 
three  adjacent  faces  equally  inclined  to  the  vertical.  Now,  as  in 
the  case  of  crystallographic  axes,  not  only  the  line  through  the 
centre  of  the  crystal,  but  any  vertical  line  will  be  an  optic  axis. 
The  principal  section  of  any  plane  of  this  crystal  is  a  vertical 
plane  perpendicular  to  it.  If  the  incident  ray  lies  in  a  principal 
section,  the  extraordinary  ray  will  lie  in  the  plane  of  incidence, 
otherwise  to  one  side  of  it.  Crystals  of  the  trimetric,  monoclinic, 
and  triclinic  system,  have  two  optic  axes  which  may  be  inclined 
at  any  angle  with  each  other.  Such  crystals  are  called  biaxial. 

When  light  slightly  inclined  to  the  optic  axis  passes  through 
the  crystal,  interference  takes  place,  producing  brightness  or  dark- 
ness according  to  the  amount  of  retardation,  or  angle  of  inclina- 


216  POLARIZED    LIGHT. 

tion.  If  rays  are  allowed  to  pass  in  all  directions  through  the 
crystal,  the  optic  axes  will  be  seen  to  be  surrounded  with  circles 
alternately  bright  and  dark,  and  colored,  owing  to  the  unequal 
wave-lengths  of  the  different  rays.  To  observe  them,  it  will  not 
do  to  lay  the  crystal  on  (7,  as  the  rays  would  then  all  be  nearly 
parallel,  but  it  must  be  held  close  to  7>,  and  inclined  from  side  to 
side.  Object  No.  1  is  a  crystal  of  Iceland  spar,  cut  perpendicular 
to  the  axis,  and  gives  readily  the  series  of  rings.  A  back  cross  is 
also  formed  with  its  centre  in  the  axis  which  changes  to  white 
when  the  analyzer  is  turned  90°.  In  observing  all  these  crystals,  it 
will  be  noticed  that  the  rings  change  only  when  they  are  inclined, 
and  not  when  moved  parallel  to  themselves,  showing  that  the  optic 
axis,  as  stated  above,  has  no  particular  position,  but  is  a  certain 
direction.  A  common  method  of  observing  the  rings  is  by 
tourmaline  tongs,  or  two  plates  of  tourmaline  with  the  crystal 
placed  between  them,  and  the  nearer  one,  or  analyzer  free  to  turn. 
Finer  effects  may  be  obtained  by  lenses  forming  a  sort  of  micro- 
scope, but  this  arrangement  is  less  simple  than  the  above.  Object 
No.  2  is  a  plate  of  quartz  cut  in  the  same  manner.  Quite  a  differ- 
ent effect  is  here  produced,  partly  because  the  retardation  is  much 
less,  unless  the  plate  is  very  thick,  and  hence  the  rings  much 
more  widely  spread,  and  partly  because  quartz  produces  what 
is  called  rotary  polarization,  that  is,  it  twists  the  plane  of  polariza- 
tion from  its  original  position  by  an  amount  depending  on  the  color, 
and  proportional  to  the  thickness.  Accordingly  the  plate  will  ap- 
pear colored,  the  tint  varying  as  the  analyzer  is  turned.  Next  try 
some  biaxial  crystals.  No.  3  is  specimen  of  arragonite,  in  which 
the  two  axes  are  visible,  surrounded  with  colored  rings,  and  with 
a  double  cross  passing  through  them,  which  changes  into  a  hyper- 
bola as  the  crystal  is  turned.  No.  4  and  5  are  crystals  of  topaz 
and  borax,  in  which  the  axes  are  so  far  separated  that  only  one 
can  be  seen  at  a  time.  The  other  may  sometimes  be  found  by 
inclining  the  crystal.  No.  6  is  a  crystal  of  nitre,  in  which  the  axes 
are  only  separated  about  5°,  and  hence  are  both  easily  seen 
together.  The  separation  of  the  rings  depends  on  the  thickness 
of  the  plate,  and  the  difference  of  the  ordinary  and  extraordinary 
indices  of  refraction,  and  is  therefore  quite  independent  of  the 
axes.  Some  crystals  give  peculiar  systems  of  rings  which  vary 


POLARISCOPE.  217 

with  each  different  specimen.  Curious  effects  may  be  obtained  by 
combining  two  or  more  plates  in  various  ways.  The  most  impor- 
tant are  the  following.  No.  7,  two  plates  of  quartz  cut  parallel  to 
the  axis,  turned  at  right  angles,  and  then  cemented  together.  A 
system  of  equilateral  hyperbolas  is  thus  obtained  with  a  common 
centre.  No.  8  is  formed  of  two  plates  cut  at  an  angle  of  45°  with 
the  axis,  and  crossed  in  the  same  way.  They  give  a  series  of 
rectilinear  bands,  forming  in  fact  the  ends  of  the  hyperbolas  of  No. 
7  with  their  asymptote,  to  which  they  are  now  parallel.  This  com- 
bination is  known  as  Savart's  plate,  and  is  important,  as  forming 
one  of  the  most  delicate  tests  for  polarized  light.  The  centre  band 
may  be  rendered  either  white  or  black  by  turning  the  analyzer 
90°.  They  are  most  intense,  when  either  parallel  or  perpendicular 
to  the  planes  of  both  analyzer  and  polarizer. 

93.    POLAKISCOPE. 

Apparatus.  A  stand  is  employed  like  a  theodolite,  or  altitude 
and  azimuth  instrument,  only  the  circles  need  be  divided  no  finer 
than  to  degrees.  The  various  forms  of  polariscopes  and  polari- 
meters  described  below,  may  be  attached  to  this,  so  that  they  are 
free  to  turn  around  their  axes,  the  angle  of  rotation  being  meas- 
ured by  a  graduated  circle  and  index.  When  the  object  to  be 
examined  is  very  minute  a  telescope  is  needed,  with  a  positive 
eye-piece,  in  which  is  a  Nicol's  prism.  In  front  of  this,  a  slide  is 
placed,  by  means  of  which  a  biquartz  or  Babinet's  wedges  may 
be  interposed  at  the  focus.  The  circle  is  attached  to  the  eye-piece, 
and  acts  like  an  eye-piece  goniometer  (p.  163).  For  common 
objects,  the  telescope  is  replaced  either  by  a  Nicol's  prism,  in  front 
of  which  a  Savart's  plate  may  be  inserted,  or  by  an  Arago's 
polariscope. 

Two  forms  of  polarimeter  are  employed,  the  first  or  common 
form,  proposed  by  Arago,  consisting  of  a  Savart's  polariscope,  in 
front  of  which  one  or  more  plates  of  glass  may  be  inserted,  and 
turned  at  any  desired  angle  so  that  the  light  may  be  more  or  less 
strongly  polarized  by  refraction.  A  graduated  circle  serves  to 
determine  the  angle  through  which  they  are  turned.  The  second 
form  of  polarimeter  consists  of  an  Arago's  polariscope,  in  which 
the  selenite  plate  is  removed,  and  a  Nicol's  prism  with  a  graduated 
circle  placed  in  front  of  the  double-image  prism,  so  that  it  may 
be  turned  through  any  desired  angle  with  regard  to  the  latter. 
These  may  also  be  mounted  *on  the  stand  like  the  polariscopes,  so 
that  they  may  be  pointed  in  any  desired  direction. 


218  POLARISCOPE. 

Experiment.  The  simplest  method  of  detecting  the  presence 
of  polarized  light  is  by  a  Nicol's  prism,  or  other  polarizer,  as 
described  in  the  last  Experiment.  Examine  in  this  way  the  light 
reflected  from  the  surface  of  the  table,  from  a  glass  plate  and 
other  sources.  Turn  the  Nicol  until  the  least  light  is  transmitted, 
and  the  direction  of  the  shorter  diagonal  of  the  face  of  the  prism 
will  give  the  plane  of  polarization.  This  method,  however,  is  not 
very  sensitive,  as  the  variation  in  intensity  of  the  light  is  not 
perceptible,  unless  a  considerable  proportion  is  polarized.  A  more 
delicate  instrument  is  the  Arago  polariscope.  This  consists  of  a 
tube  with  a  square  aperture  at  one  end,  and  a  double-image  prism 
and  plate  of  selenite  at  the  other.  The  size  of  the  aperture  is 
such,  that  the  two  images  of  it  shall  be  just  in  contact,  but  not 
overlapping.  If,  now,  a  ray  of  polarized  light  is  viewed  through 
the  prism,  the  two  images  will  in  general  assume  complementary 
colors.  When  the  line  of  separation  of  the  two  images  is 
parallel  or  perpendicular  to  the  plane  of  polarization,  the  colors 
are  most  strongly  marked,  and  they  disappear  when  the  angle  of 
inclination  is  45°.  To  determine  the  plane  of  polarization  of  any 
ray,  first  direct  the  instrument  towards  the  light  reflected  from 
a  polished  horizontal  surface,  turn  the  line  of  separation  of  the 
two  images  until  it  is  vertical,  or  parallel  to  the  plane  of  polar- 
ization, and  note  the  color  of  the  right  hand  image.  Now  direct 
the  tube  towards  the  source  of  light  and  turn  it  until  this  image 
has  the  same  color  as  before.  The  plane  of  polarization  is  then 
parallel  to  the  line  of  separation.  It  is  well  to  make  a  notch  in 
one  side  of  the  square,  which  will  then  appear  in  a  different  part 
of  the  two  images  and  thus  serve  to  distinguish  them.  The  exact 
direction  of  the  plane  of  polarization  may  be  found  by  noting 
when  the  colors  are  most  marked,  or,  more  accurately,  by  bisect- 
ing the  two  positions  where  they  disappear,  each  of  which  is  45° 
distant  from  it.  The  delicacy  of  thi§  instrument  is  much  greater 
than  that  of  a  simple  Nicol's  prism,  as  with  it  about  three  or  four 
per  cent,  of  polarized  light  can  be  detected. 

Sometimes  the  Arago  polariscope  is  used  without  the  tube. 
For  instance,  in  observing  the  polarization  of  the  solar  corona 
during  a  total  eclipse,  doubt  was  cast  on  the  results  by  the  strong 
polarization  of  the  sky.  To  eliminate  this,  the  tube  was  removed, 


POLARISCOPE.  219 

in  which  ease  the  two  images  of  the  sky  overlapped,  producing 
unpolarized  light,  while  the  images  of  the  corona  were  separated 
so  that  they  appeared  on  a  white  unpolarized  back-ground. 

A  still  more  delicate  form  of  polariscope  is  that  proposed  by 
Savart,  which  consists  of  a  Nicol's  piism  with  a  double  plate  of 
quartz,  giving  bands  as  described  in  Experiment  92,  specimen  No. 
8.  The  plate  is  attached  to  the  Nicol  so  that  the  bands  shall  be 
perpendicular  to  its  principal  plane,  in  which  case,  when  parallel 
to  the  plane  of  polarization  they  will  be  black-centred,  and  when 
perpendicular  to  it,  white-centred.  If  the  bands  were  parallel  to 
the  plane  of  the  Nicol,  this  effect  would  be  reversed.  It  is  now 
very  easy  to  determine  the  plane  of  polarization  of  a  given  ray. 
The  instrument  is  turned  until  the  bands  are  black-centred,  when 
their  direction  marks  that  of  the  plane.  The  position  of  the  latter 
is  then  found  more  precisely  by  bisecting  the  two  points  of  disap- 
pearance of  the  bands.  This  instrument  is  more  sensitive  than 
either  of  the  preceding,  as  by  it  one  or  two  per  cent,  of  polarized 
light  can  be  detected.  Try  these  different  instruments  on  various 
sources  of  polarized  light,  and  see  if  all  give  the  same  results  for 
the  direction  of  the  plane  of  polarization.  For  instance,  see  if  the 
light  reflected  by  paper,  wood  or  cloth  is  polarized  in  the  plane 
of  incidence,  and  if  that  transmitted  obliquely  through  glass  is 
polarized  in  a  plane  perpendicular  to  the  plane  of  refraction. 

Now  direct  the  telescope  towards  some  source  of  polarized  light, 
and  observe  its  plane  with  the  simple  Nicol's  prism.  Then  push 
the  slide  so  as  to  interpose  the  biquartz,  which  consists  of  two 
pieces  of  quartz  joined  together,  one  turning  the  ray  to  the  right, 
the  other  to  the  left.  The  two  halves  will  then  assume  comple- 
mentary colors,  unless  the  plane  of  polarization  is  parallel  or  per- 
pendicular to  their  line  of  junction.  In  the  first  case,  the  color 
of  both  is  a  sort  of  pale  violet,  in  the  second,  yellowish  brown. 
Make  a  number  of  observations  of  the  angle  of  the  plane,  and 
compute  the  probable  error.  Now  try  the  effect  of  the  other 
quartz  plate.  This  is  composed  of  two  wedges  of  quartz  cemented 
together,  one  turning  the  ray  to  the  right;  the  other  to  the  left. 
Accordingly  a  series  of  bands  are  produced,  which  disappear  when 
parallel  or  perpendicular  to  the  plane  of  polarization.  Repeat  the 
observations  with  this  plate,  and  compare  its  probable  error  with 


220  POLARISCOPE. 

that  of  the  biquartz.  It  will  be  noticed  that  both  these  devices 
require  a  parallel  beam,  while  the  Savart's  polariscope,  which  needs 
a  converging  beam,  cannot  be  attached  to  a  telescope  in  this  way, 
but  must  be  placed  in  front  of  the  eye-piece.  In  this  case  it  cuts 
down  the  field  of  view,  and  is  therefore  inconvenient  to  use. 

To  measure  the  proportion  of  polarized  light  in  a  given  beam, 
the  polarimeter  is  employed.  This  consists  of  a  Savart,  or  other 
form  of  polariscope,  in  front  of  which  some  plates  of  glass  are  placed, 
free  to  turn,  so  that  the  transmitted  light  may  pass  through 
them  at  any  desired  angle.  It  will  thus  be  polarized  by  refrac- 
tion to  a  greater  or  less  extent,  depending  on  the  number  of  plates, 
and  the  angle  through  which  they  are  turned.  To  measure  this, 
a  graduated  circle  is  attached,  which  may  be  divided  either  into 
degrees,  or  so  as  to  give  the  percentage  directly.  The  bands 
should  be  placed  parallel  to  the  axis  around  which  the  plates  turn. 
To  use  this  instrument,  set  the  plates  at  0°  and  direct  it  towards  a 
source  of  unpolarized  light.  The  field  will  now  be  perfectly  uniform. 
Turn  the  plates,  and  the  bands  will  appear  faintly,  dark-centred 
and  increasing  in  strength  with  the  angle.  Turn  the  plates  back 
to  0°,  and  direct  the  instrument  towards  the  light  to  be  examined, 
find  its  plane  of  polarization  and  bring  the  bands  to  a  position  at 
right  angles  to  it,  that  is,  so  that  they  shall  be  most  strongly  light- 
centred.  ISTow  on  turning  the  plates,  they  tend  to  neutralize  the 
polarization,  since  they  tend  to  polarize  it  in  a  plane  passing 
through  this  axis,  while  it  is  already  polarized  in  a  plane  perpen- 
dicular to  this.  As  they  are  turned,  the  bands  therefore  become 
fainter  and  fainter,  then  disappear  and  reappear  dark-centred, 
when  the  angle  becomes  too  great.  At  the  point  of  disappearance, 
the  polarization  produced  by  the  plates  is  just  equal  to  that  already 
present  in  the  beam,  the  transmitted  light  is  therefore  unpolarized, 
and  gives  no  bands.  Take  a  number  of  readings  of  the  point  of 
disappearance,  first  turning  the  plates  to  the  right,  and  then  to  the 
left,  and  reduce  to  percentages  by  means  of  a  table  which  should 
accompany  the  instrument. 

The  difficulty  of  computing  this  table  with  accuracy,  greatly 
diminishes  the  value  of  this  instrument.  The  theoretical  formulas 
are  quite  complex,  and  of  little  use  on  account  of  the  difficulty  of 
allowing  for  the  light  absorbed  by  the  glass.  It  must  therefore  be 


POLARISCOPE.  221 

determined  experimentally  by  observing  with  it  a  beam  in  which 
the  percentage  of  polarized  light  may  be  regulated  at  will.  This 
may  be  accomplished  either  by  setting  a  plate  of  glass  at  an  angle 
of  45°,  and  varying  the  relative  intensities  of  the  reflected  and  re- 
fracted beams,  or  by  reuniting  two  beams  of  variable  intensity  by 
means  of  a  double-image  prism.  Again,  if  the  beam  is  strongly 
polarized  it  is  impossible  to  make  fche  bands  disappear,  unless  a 
large  number  of  plates  are  used,  in  which  case  the  transmitted 
beam  is  very  feeble. 

These  various  difficulties  are  obviated  by  the  other  form  of 
polarimeter.  As  in  the  Arago,  two  adjacent  images  of  the  square 
are  formed,  one  polarized  horizontally,  the  other  vertically,  which 
will  have  equal  intensities  if  the  light  is  unpolarized,  but  one  of 
which  will  be  in  general,  brighter  than  the  other,  when  viewed  by 
polarized  light.  If  now  the  two  images  are  seen  through  a  Nicol's 
prism,  their  relative  intensities  will  vary  as  it  is  turned,  each  dis- 
appearing when  the  plane  of  the  Nicol  is  perpendicular  to  its  own. 
Accordingly,  certain  positions  can  always  be  found,  in  which  the 
two  images  will  have  precisely  the  same  brightness,  and  the  angle 
through  which  the  Nicol  has  been  turned,  gives  a  measure  of  their 
true  relative  intensities,  and  hence  the  percentage  of  polarized 
light  present.  To  make  the  reduction,  call  a  the  angle  through 
which  the  Nicol  has  been  turned,  A  the  amount  of  light  polarized 
in  a  vertical  plane,  and  _Z?  that  polarized  horizon- 
tally. Thus  if  the  plane  is  vertical,  A  is  greater 
than  .Z?,  and  A  —  B  is  the  amount  of  free  polarized 
light.  A  +  J3  being  the  total  intensity  of  the 
light,  their  ratio  gives  the  percentage  of  polariza- 
tion. When  the  plane  of  the  Nicol  is  vertical,  A 
retains  its  full  brilliancy,  which,  at  any  other  angle 
is  reduced  in  the  ratio  cos2  a.  JS  is  in  like  manner 
proportional  to  sin2  a.  The  percentage  of  polarized 

A  —  B        cos2  a  —  sin2  a 

light  n  therefore  equals  —A — j — ^  =  « ; — :— 5 — 

A  ~r  -B         cos2  a  -f-  sin2  a 

=  cos2  a  —  sin2  a  =  cos  2a.  The  reduction  may  then  be  effected 
by  a  table  of  natural  cosines,  or  by  the  accompanying  table  which 
gives  the  corresponding  values  of  a  and  n. 

To  use  this  instrument  direct  it  towards  the  source  of  light  to 


0  100.0 

5  98.5 

10  94.0 

15  86.6 

20  76.6 

25  64.3 

30  50.0 

35  34.2 

40  17.4 

4o  0 


222  SACCHARIMETER. 

be  examined,  and  turn  it  so  that  the  line  separating  the  two 
squares  shall  be  parallel  to  the  plane  of  polarization.  Then  turn 
the  Nicol  until  the  two  images  are  equally  bright,  when  the  angle 
will  give,  by  means  of  the  table>  the  percentage  of  polarized  light 
present.  It  is  best  to  take  readings  on  each  side  of  the  0°  and 
employ  the  mean,  thus  eliminating  any  error  in  the  0°  point. 
Otherwise,  care  must  be  taken  that  the  circle  is  fixed  in  such  a 
position  that  when  the  Nicol  is  turned  so  that  one  image  shall 
completely  disappear,  the  reading  of  the  index  shall  be  precisely  0° 
or  90°. 

Now  measure  with  the  polarimeters  the  amount  of  polarized 
light  contained  in  the  rays  whose  plane  of  polarization  was  pre- 
viously determined.  Next  throw  a  beam  of  sunlight  upon  a 
sheet  of  paper,  and  measure  the  percentage  of  polarization  of  the 
light  thrown  out  in  various  directions.  Observations  of  this  kind 
are  much  needed  for  various  substances  at  different  angles  of 
incidence  and  reflection.  It  will  be  found  that  it  is  extremely 
difficult  to  obtain  a  beam  from  a  large  surface  entirely  free  from 
all  traces  of  polarization,  and  hence  much  care  is  needed  to  ob- 
tain really  accurate  results. 

When  the  sky  is  clear  its  light  is  found  to  be  strongly  polarized 
in  planes  passing  through  the  sun,  the  effect  being  most  marked, 
at  a  distance  of  90°  from  that  body.  Beyond  90°  the  polarization 
again  diminishes,  and  becomes  zero  at  a  point  called  the  neutral 
point  in  the  same  vertical  plane  as  the  sun,  but  150°  distant, 
below  this  point  the  plane  of  polarization  becomes  horizontal. 
Two  other  neutral  points  exist,  one  17°  below  the  sun,  the  other 
8£°  above  it,  -but  both  much  more  difficult  to  observe.  Even  a 
faint  cloud  alters  these  effects,  and  when  the  sky  is  entirely 
covered  with  clouds,  no  polarization  is  perceptible.  Very  valuable 
work  might  be  done  by  measuring  the  plane  and  amount  of  polar- 
ization of  the  light  in  different  parts  of  the  sky. 

94.      SACCHARIMETER. 

Apparatus.  A  Soleil  saccharimeter,  some  pure  sugar  and  some 
unrefined,  or  brown  sugar.  Also  some  chlorhydric  acid  and  sub- 
acetate  of  lead,  a  flask  containing  just  100  cm.8,  a  funnel,  filter 
paper,  a  balance  and  weights. 


SACCHARIMETER.  223 

Experiment.  The  most  important  practical  application  of  po- 
larized light  is  to  sacchariraetry,  or  the  measurement  of  the 
strength  of  a  solution  of  sugar.  This  depends  on  the  property  of 
such  a  solution  of  producing  rotary  polarization,  or  of  turning  the 
plane  of  a  beam  of  polarized  light  by  an  amount  proportional  to 
the  amount  of  sugar  present.  The  saccharimeter  is  merely  an 
instrument  for  measuring  the  angular  change  of  the  plane,  or  more 
strictly,  the  thickness  of  a  plate  of  quartz,  rotating  it  in  the  oppo- 
site direction,  required  to  bring  it  back  to  its  primitive  position. 
The  liquid  is  contained  in  a  brass  tube  closed  at  each  end  with 
plates  of  glass,  which  are  held  in  place  by  screw  caps.  The  light 
first  passes  through  a  circular  aperture,  two  polarized  images  of 
which  are  formed  by  a  double-image  prism,  and  one  transmitted 
through  the  instrument,  the  other  thrown  off  to  one  side.  It 
next  passes  through  a  double  plate  of  quartz  formed  of  two  semi- 
circles, one  of  which  turns  the  ray  to  the  right,  the  other  to  the 
left.  It  next  passes  through  the  column  of  sugar  by  which  both 
rays  are  turned,  by  a  certain  amount,  to  the  right.  It  is  brought 
back  to  its  primitive  position  by  a  compensater,  formed  of  a  plate 
and  two  wedges  of  quartz,  the  latter  being  turned  in  opposite 
directions,  and  carrying  racks  which  are  acted  on  by  a  pinion  so 
that  they  may  be  moved  past  each  other  by  any  desired  amount. 
The  thickness  of  the  layer  of  quartz  may  thus  be  varied  at  will, 
and  accurately  determined  by  a  scale  attached  to  one  wedge,  and 
an  index  to  the  other.  The  light  next  passes  through  a  Nicol's 
prism  which  serves  as  an  analyzer,  and  then  through  a  small  Gal- 
ilean telescope  by  which  an  enlarged  image  of  the  biquartz  is 
formed.  In  front  of  the  eye-piece  of  the  telescope  is  an  additional 
Nicol's  prism  and  plate  of  quartz,  the  latter  being  free  to  turn. 
The  object  of  this  is  to  vary  the  tint  of  the  two  halves  of  the  bi- 
quartz, so  that  the  color  to  which  the  eye  is  most  sensitive  may  be 
selected,  and  also  to  neutralize  any  color  already  present  in  the 
solution. 

To  use  this  instrument,  weigh  out  16.47  grammes  of  pure  sugar 
and  dissolve  it  in  enough  water  to  make  the  solution  occupy  100 
cm8.  Unscrew  the  cap  from  one  of  the  tubes,  fill  it  with  water 
and  slide  on  the  glass  plate,  taking  great  care  that  no  air-bubbles 
are  imprisoned  under  it.  Replace  the  cap  and  wipe  the  exterior 


224 


SACCHARIMETER. 


dry.  Fill  a  second  tube  in  the  same  manner  with  the  solution  of 
sugar.  Turn  the  stand  towards  the  light,  lay  the  tube  containing 
water  in  place  on  it,  and  focus  the  telescope  on  the  biquartz  by 
drawing  out  the  eye-piece  until  the  line  of  separation  is  distinctly 
visible.  The  two  semicircles  will  now,  in  general,  appear  of  differ- 
ent colors  which  may  be  changed  by  moving  the  wedges  by  the 
milled  head  below.  A  certain  position  will  be  found,  however,  in 
which  they  are  alike,  and  the  reading  of  the  scale  should  then  be 
zero. 

When  the  two  halves  appear  of  the  same  tint,  turn  the  quartz 
plate  in  the  eye-piece,  by  which  their  color  is  altered.  There  is  a 
peculiar  purplish  brown  color,  different  for  different  eyes,  from 
which  the  two  halves  change  more  rapidly  than  from  any  other, 
when  the  wedges  are  moved.  Consequently,  when  of  this 
color,  which  is  called  the  sensitive  tint,  they  can  be  set  more 
accurately  than  in  any  other  case.  To  obtain  this  sensitive  tint 
bring  the  two  halves  as  nearly  alike  as  possible,  then  turn  the 
quartz  and  see  if  any  difference  is  perceptible ;  if  so,  set  again,  un- 
til no  difference  can  be  detected  in  the  two  halves,  however  the 
the  plate  is  turned.  Take  a  number  of  observations,  and  reading 
the  scale  to  tenths  of  a  division,  take  the  mean.  If  not  zero,  it 
may  be  brought  to  this  position  by  means  of  a  small  screw,  which 
moves  the  scale  without  affecting  the  wedges. 

Now  replace  the  tube  containing  water  by  that  containing  a 
solution  of  sugar,  when  it  will  be  found  that  the  semicircles 
have  very  different  colors,  and  on  making  them  alike,  the  reading 
of  the  scale  becomes  100,  if  the  sugar  is  perfectly  pure.  As  before, 
take  the  mean  of  several  readings,  and  turn  the  quartz  each  time 
to  obtain  the  most  sensitive  tint. 

Next  make  a  solution  of  one  half  the  strength  by  mixing  some 
of  the  standard  solution  with  exactly  its  own  volume  of  water,  and 
see  if  the  reading  is  50.  Then  dilute  again  one  half,  to  get  a 
solution  of  strength  one  fourth,  and  see  if  the  reading  is  25.  If 
kept  for  some  time,  the  solution  will  ferment,  and  the  reading 
diminish,  especially  during  warm  weather,  or  if  exposed  to  the  air. 
In  general,  a  solution  of  impure  sugar  is  not  transparent,  and  is 
often  so  opaque,  that  the  semicircles  cannot  be  observed  through 
it.  In  this  case  it  must  be  clarified  by  adding  some  sub-acetate  of 


SACCHARIMETER.  225 

lead  and  then  filtering.  Animal  charcoal  was  at  one  time  used, 
but  it  is  found  that  this  absorbs  some  of  the  sugar  with  the  im- 
purities. In  practice  the  problem  generally  is  complicated  by  the 
fact,  that  the  molasses  and  other  impurities  commonly  found  in 
sugar,  also  turn  the  plane  of  polarization  to  the  right,  and  thus 
render  the  results  uncertain.  This  effect  must  therefore  be  elimi- 
nated by  adding  to  the  solution  one  tenth  of  its  bulk  of  pure 
chlorhydric  acid,  and  heating  to  68°  C.  The  cane-sugar  is  thus 
converted  into  grape-sugar,  which  turns  the  plane  of  polarization 
by  an  equal  amount  to  the  left,  while  it  does  not  effect  the  mo- 
lasses and  uncrystallizable  sugar.  After  heating,  the  solution  is 
poured  into  the  larger  tube,  which  has  an  aperture  in  one  side  to 
contain  a  thermometer.  The  length  of  the  column  in  this  case 
is  one  tenth  greater  than  before,  which  just  compensates  for  the 
dilution  due  to  the  addition  of  the  chlorhydric  acid.  The  reading 
should  then  be  taken,  and  the  temperature  noted.  As  this  read- 
ing gives  the  difference  of  the  amount  of  crystallizable  and  un- 
crystallizable sugar,  and  the  first  reading  gives  their  sum,  the 
amount  of  crystallizable  sugar  may  be  obtained  by  taking  half  the 
sum  of  the  two  readings,  and  the  amount  of  uncrystallizable,  by 
taking  half  their  difference.  Thus  if  this  first  reading  is  80  and 
the  second  30,  it  denotes  that  there  is  55  per  cent,  of  crystalliza- 
ble, and  25  per  cent  of  syrup.  A  correction  must  be  applied  for 
temperature,  which  is  best  done  by  means  of  a  table,  which  ac- 
companies the  instrument. 

15 


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